HomeHome Intuitionistic Logic Explorer
Theorem List (p. 29 of 102)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelrabsf 2801 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2696 has implicit substitution). The hypothesis specifies that 
x must not be a free variable in  B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x B   =>    |-  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ( A  e.  B  /\  [. A  /  x ].
 ph ) )
 
Theoremeqsbc3 2802* Substitution applied to an atomic wff. Set theory version of eqsb3 2141. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
 
Theoremsbcng 2803 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ].
 ph ) )
 
Theoremsbcimg 2804 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps )  <->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps ) ) )
 
Theoremsbcan 2805 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 )
 
Theoremsbcang 2806 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 ) )
 
Theoremsbcor 2807 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
 
Theoremsbcorg 2808 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
 
Theoremsbcbig 2809 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps ) ) )
 
Theoremsbcal 2810* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph )
 
Theoremsbcalg 2811* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
 
Theoremsbcex2 2812* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
 |-  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph )
 
Theoremsbcexg 2813* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
 
Theoremsbceqal 2814* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
 |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B )
 )
 
Theoremsbeqalb 2815* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
 |-  ( A  e.  V  ->  ( ( A. x ( ph  <->  x  =  A )  /\  A. x (
 ph 
 <->  x  =  B ) )  ->  A  =  B ) )
 
Theoremsbcbid 2816 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. A  /  x ].
 ch ) )
 
Theoremsbcbidv 2817* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. A  /  x ].
 ch ) )
 
Theoremsbcbii 2818 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
 |-  ( ph  <->  ps )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps )
 
Theoremeqsbc3r 2819* eqsbc3 2802 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A )
 )
 
Theoremsbc3ang 2820 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps  /\ 
 ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) ) )
 
Theoremsbcel1gv 2821* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
 
Theoremsbcel2gv 2822* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
 
Theoremsbcimdv 2823* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\  A  e.  V ) 
 ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
 )
 
Theoremsbctt 2824 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/ x ph )  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbcgf 2825 Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 F/ x ph   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc19.21g 2826 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
 |- 
 F/ x ph   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps )  <->  ( ph  ->  [. A  /  x ]. ps ) ) )
 
Theoremsbcg 2827* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2825. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc2iegf 2828* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |- 
 F/ x ps   &    |-  F/ y ps   &    |-  F/ x  B  e.  W   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( [. A  /  x ].
 [. B  /  y ]. ph  <->  ps ) )
 
Theoremsbc2ie 2829* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
 
Theoremsbc2iedv 2830* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [. A  /  x ]. [. B  /  y ]. ps  <->  ch ) )
 
Theoremsbc3ie 2831* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  (
 ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps ) )   =>    |-  ( [. A  /  x ]. [. B  /  y ]. [. C  /  z ]. ph  <->  ps )
 
Theoremsbccomlem 2832* Lemma for sbccom 2833. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom 2833* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbcralt 2834* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
 |-  ( ( A  e.  V  /\  F/_ y A ) 
 ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcrext 2835* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( F/_ y A  ->  (
 [. A  /  x ].
 E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
 )
 
Theoremsbcralg 2836* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcrex 2837* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)
 |-  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph )
 
Theoremsbcreug 2838* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcabel 2839* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
 |-  F/_ x B   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  | 
 ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B ) )
 
Theoremrspsbc 2840* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 1658 and spsbc 2775. See also rspsbca 2841 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  [. A  /  x ]. ph )
 )
 
Theoremrspsbca 2841* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  [. A  /  x ]. ph )
 
Theoremrspesbca 2842* Existence form of rspsbca 2841. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( ( A  e.  B  /\  [. A  /  x ].
 ph )  ->  E. x  e.  B  ph )
 
Theoremspesbc 2843 Existence form of spsbc 2775. (Contributed by Mario Carneiro, 18-Nov-2016.)
 |-  ( [. A  /  x ]. ph  ->  E. x ph )
 
Theoremspesbcd 2844 form of spsbc 2775. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  [. A  /  x ]. ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbcth2 2845* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( x  e.  B  -> 
 ph )   =>    |-  ( A  e.  B  -> 
 [. A  /  x ].
 ph )
 
Theoremra5 2846 Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1476. (Contributed by NM, 16-Jan-2004.)
 |- 
 F/ x ph   =>    |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  A. x  e.  A  ps ) )
 
Theoremrmo2ilem 2847* Condition implying restricted "at most one." (Contributed by Jim Kingdon, 14-Jul-2018.)
 |- 
 F/ y ph   =>    |-  ( E. y A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A  ph )
 
Theoremrmo2i 2848* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E. y  e.  A  A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A  ph )
 
Theoremrmo3 2849* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E* x  e.  A  ph  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
Theoremrmob 2850* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  C  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( E* x  e.  A  ph  /\  ( B  e.  A  /\  ps ) )  ->  ( B  =  C  <->  ( C  e.  A  /\  ch ) ) )
 
Theoremrmoi 2851* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  C  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( E* x  e.  A  ph  /\  ( B  e.  A  /\  ps )  /\  ( C  e.  A  /\  ch ) )  ->  B  =  C )
 
2.1.10  Proper substitution of classes for sets into classes
 
Syntaxcsb 2852 Extend class notation to include the proper substitution of a class for a set into another class.
 class  [_ A  /  x ]_ B
 
Definitiondf-csb 2853* Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 2764, to prevent ambiguity. Theorem sbcel1g 2869 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 2878 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
 
Theoremcsb2 2854* Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that  x can be free in  B but cannot occur in  A. (Contributed by NM, 2-Dec-2013.)
 |-  [_ A  /  x ]_ B  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
 
Theoremcsbeq1 2855 Analog of dfsbcq 2766 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  =  B  -> 
 [_ A  /  x ]_ C  =  [_ B  /  x ]_ C )
 
Theoremcbvcsb 2856 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on  A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ y C   &    |-  F/_ x D   &    |-  ( x  =  y  ->  C  =  D )   =>    |-  [_ A  /  x ]_ C  =  [_ A  /  y ]_ D
 
Theoremcbvcsbv 2857* Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( x  =  y 
 ->  B  =  C )   =>    |-  [_ A  /  x ]_ B  =  [_ A  /  y ]_ C
 
Theoremcsbeq1d 2858 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  [_ A  /  x ]_ C  =  [_ B  /  x ]_ C )
 
Theoremcsbid 2859 Analog of sbid 1657 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ x  /  x ]_ A  =  A
 
Theoremcsbeq1a 2860 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
 
Theoremcsbco 2861* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  [_ A  /  x ]_ B
 
Theoremcsbtt 2862 Substitution doesn't affect a constant  B (in which  x is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/_ x B ) 
 ->  [_ A  /  x ]_ B  =  B )
 
Theoremcsbconstgf 2863 Substitution doesn't affect a constant  B (in which  x is not free). (Contributed by NM, 10-Nov-2005.)
 |-  F/_ x B   =>    |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  B )
 
Theoremcsbconstg 2864* Substitution doesn't affect a constant  B (in which  x is not free). csbconstgf 2863 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  B )
 
Theoremsbcel12g 2865 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
 
Theoremsbceqg 2866 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
 
Theoremsbcnel12g 2867 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e/  C  <->  [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C ) )
 
Theoremsbcne12g 2868 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
 
Theoremsbcel1g 2869* Move proper substitution in and out of a membership relation. Note that the scope of  [. A  /  x ]. is the wff  B  e.  C, whereas the scope of  [_ A  /  x ]_ is the class  B. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  C )
 )
 
Theoremsbceq1g 2870* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  C )
 )
 
Theoremsbcel2g 2871* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  B  e.  [_ A  /  x ]_ C ) )
 
Theoremsbceq2g 2872* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  B  =  [_ A  /  x ]_ C ) )
 
Theoremcsbcomg 2873* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  [_ A  /  x ]_
 [_ B  /  y ]_ C  =  [_ B  /  y ]_ [_ A  /  x ]_ C )
 
Theoremcsbeq2d 2874 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |- 
 F/ x ph   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
 
Theoremcsbeq2dv 2875* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C )
 
Theoremcsbeq2i 2876 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)
 |-  B  =  C   =>    |-  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C
 
Theoremcsbvarg 2877 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ x  =  A )
 
Theoremsbccsbg 2878* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  y  e.  [_ A  /  x ]_ { y  |  ph } ) )
 
Theoremsbccsb2g 2879 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A  e.  [_ A  /  x ]_ { x  |  ph } ) )
 
Theoremnfcsb1d 2880 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x [_ A  /  x ]_ B )
 
Theoremnfcsb1 2881 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x [_ A  /  x ]_ B
 
Theoremnfcsb1v 2882* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x [_ A  /  x ]_ B
 
Theoremnfcsbd 2883 Deduction version of nfcsb 2884. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x [_ A  /  y ]_ B )
 
Theoremnfcsb 2884 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x [_ A  /  y ]_ B
 
Theoremcsbhypf 2885* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2603 for class substitution version. (Contributed by NM, 19-Dec-2008.)
 |-  F/_ x A   &    |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( y  =  A  ->  [_ y  /  x ]_ B  =  C )
 
Theoremcsbiebt 2886* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2890.) (Contributed by NM, 11-Nov-2005.)
 |-  ( ( A  e.  V  /\  F/_ x C ) 
 ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
 
Theoremcsbiedf 2887* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/_ x C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A )  ->  B  =  C )   =>    |-  ( ph  ->  [_ A  /  x ]_ B  =  C )
 
Theoremcsbieb 2888* Bidirectional conversion between an implicit class substitution hypothesis  x  =  A  ->  B  =  C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)
 |-  A  e.  _V   &    |-  F/_ x C   =>    |-  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C )
 
Theoremcsbiebg 2889* Bidirectional conversion between an implicit class substitution hypothesis  x  =  A  ->  B  =  C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x C   =>    |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  B  =  C )  <->  [_ A  /  x ]_ B  =  C ) )
 
Theoremcsbiegf 2890* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( A  e.  V  -> 
 F/_ x C )   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  C )
 
Theoremcsbief 2891* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  A  e.  _V   &    |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  [_ A  /  x ]_ B  =  C
 
Theoremcsbied 2892* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  B  =  C )   =>    |-  ( ph  ->  [_ A  /  x ]_ B  =  C )
 
Theoremcsbied2 2893* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A  =  B )   &    |-  (
 ( ph  /\  x  =  B )  ->  C  =  D )   =>    |-  ( ph  ->  [_ A  /  x ]_ C  =  D )
 
Theoremcsbie2t 2894* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2895). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A. x A. y ( ( x  =  A  /\  y  =  B )  ->  C  =  D )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  D )
 
Theoremcsbie2 2895* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  C  =  D )   =>    |-  [_ A  /  x ]_
 [_ B  /  y ]_ C  =  D
 
Theoremcsbie2g 2896* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2797 avoids a disjointness condition on  x and  A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
 |-  ( x  =  y 
 ->  B  =  C )   &    |-  ( y  =  A  ->  C  =  D )   =>    |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  D )
 
Theoremsbcnestgf 2897 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
 |-  ( ( A  e.  V  /\  A. y F/ x ph )  ->  ( [. A  /  x ].
 [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
 
Theoremcsbnestgf 2898 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
 |-  ( ( A  e.  V  /\  A. y F/_ x C )  ->  [_ A  /  x ]_ [_ B  /  y ]_ C  =  [_
 [_ A  /  x ]_ B  /  y ]_ C )
 
Theoremsbcnestg 2899* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. ph ) )
 
Theoremcsbnestg 2900* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_
 [_ B  /  y ]_ C  =  [_ [_ A  /  x ]_ B  /  y ]_ C )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10124
  Copyright terms: Public domain < Previous  Next >