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Theorem ralprg 3421
Description: Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ralprg.2  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
ralprg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
Distinct variable groups:    x, A    x, B    ps, x    ch, x
Allowed substitution hints:    ph( x)    V( x)    W( x)

Proof of Theorem ralprg
StepHypRef Expression
1 df-pr 3382 . . . 4  |-  { A ,  B }  =  ( { A }  u.  { B } )
21raleqi 2509 . . 3  |-  ( A. x  e.  { A ,  B } ph  <->  A. x  e.  ( { A }  u.  { B } )
ph )
3 ralunb 3124 . . 3  |-  ( A. x  e.  ( { A }  u.  { B } ) ph  <->  ( A. x  e.  { A } ph  /\  A. x  e.  { B } ph ) )
42, 3bitri 173 . 2  |-  ( A. x  e.  { A ,  B } ph  <->  ( A. x  e.  { A } ph  /\  A. x  e.  { B } ph ) )
5 ralprg.1 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
65ralsng 3411 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  ps ) )
7 ralprg.2 . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
87ralsng 3411 . . 3  |-  ( B  e.  W  ->  ( A. x  e.  { B } ph  <->  ch ) )
96, 8bi2anan9 538 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A. x  e.  { A } ph  /\ 
A. x  e.  { B } ph )  <->  ( ps  /\ 
ch ) ) )
104, 9syl5bb 181 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A. x  e. 
{ A ,  B } ph  <->  ( ps  /\  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306    u. cun 2915   {csn 3375   {cpr 3376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382
This theorem is referenced by:  raltpg  3423  ralpr  3425  iinxprg  3731  fvinim0ffz  9094
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