Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ax11wK Unicode version

Theorem ax11wK 28174
Description: Weak version of ax-11 1624 from which we can prove any ax-7 1535 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes (see description for equidK 28136). An instance of the first hypothesis will normally require that  x and  y be distinct (unless  x does not occur in  ph). See the description in the comment of equidK 28136. (Contributed by NM, 10-Apr-2017.)
Hypotheses
Ref Expression
ax11wK.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
ax11wK.2  |-  ( y  =  z  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
ax11wK  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Distinct variable groups:    y, z    ps, x    ph, z    ch, y
Allowed substitution hints:    ph( x, y)    ps( y, z)    ch( x, z)

Proof of Theorem ax11wK
StepHypRef Expression
1 ax11wK.2 . . 3  |-  ( y  =  z  ->  ( ph 
<->  ch ) )
21ax4wK 28153 . 2  |-  ( A. y ph  ->  ph )
3 ax11wK.1 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43ax11wlemK 28173 . 2  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
52, 4syl5 30 1  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532
This theorem is referenced by:  ax11wdemoK  28176
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-8 1623  ax-17 1628  ax-9v 1632
This theorem depends on definitions:  df-bi 179  df-an 362
  Copyright terms: Public domain W3C validator