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Theorem cbvral2 27456
Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 2857. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Hypotheses
Ref Expression
cbvral2.1  |-  F/ z
ph
cbvral2.2  |-  F/ x ch
cbvral2.3  |-  F/ w ch
cbvral2.4  |-  F/ y ps
cbvral2.5  |-  ( x  =  z  ->  ( ph 
<->  ch ) )
cbvral2.6  |-  ( y  =  w  ->  ( ch 
<->  ps ) )
Assertion
Ref Expression
cbvral2  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
Distinct variable groups:    x, A    z, A    x, y, B   
y, z, B    w, B
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z, w)    ch( x, y, z, w)    A( y, w)

Proof of Theorem cbvral2
StepHypRef Expression
1 nfcv 2502 . . . 4  |-  F/_ z B
2 cbvral2.1 . . . 4  |-  F/ z
ph
31, 2nfral 2681 . . 3  |-  F/ z A. y  e.  B  ph
4 nfcv 2502 . . . 4  |-  F/_ x B
5 cbvral2.2 . . . 4  |-  F/ x ch
64, 5nfral 2681 . . 3  |-  F/ x A. y  e.  B  ch
7 cbvral2.5 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  ch ) )
87ralbidv 2648 . . 3  |-  ( x  =  z  ->  ( A. y  e.  B  ph  <->  A. y  e.  B  ch ) )
93, 6, 8cbvral 2845 . 2  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. y  e.  B  ch )
10 cbvral2.3 . . . 4  |-  F/ w ch
11 cbvral2.4 . . . 4  |-  F/ y ps
12 cbvral2.6 . . . 4  |-  ( y  =  w  ->  ( ch 
<->  ps ) )
1310, 11, 12cbvral 2845 . . 3  |-  ( A. y  e.  B  ch  <->  A. w  e.  B  ps )
1413ralbii 2652 . 2  |-  ( A. z  e.  A  A. y  e.  B  ch  <->  A. z  e.  A  A. w  e.  B  ps )
159, 14bitri 240 1  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1549   A.wral 2628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633
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