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Theorem cbvral2vw 2714
Description: Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2716 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
cbvral2vw.1  |-  ( x  =  z  ->  ( ph 
<->  ch ) )
cbvral2vw.2  |-  ( y  =  w  ->  ( ch 
<->  ps ) )
Assertion
Ref Expression
cbvral2vw  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
Distinct variable groups:    x, z    y, w    x, A, z    x, y, B, z    w, B    ph, z    ps, y    ch, x    ch, w
Allowed substitution hints:    ph( x, y, w)    ps( x, z, w)    ch( y, z)    A( y, w)

Proof of Theorem cbvral2vw
StepHypRef Expression
1 cbvral2vw.1 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  ch ) )
21ralbidv 2477 . . 3  |-  ( x  =  z  ->  ( A. y  e.  B  ph  <->  A. y  e.  B  ch ) )
32cbvralvw 2707 . 2  |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. y  e.  B  ch )
4 cbvral2vw.2 . . . 4  |-  ( y  =  w  ->  ( ch 
<->  ps ) )
54cbvralvw 2707 . . 3  |-  ( A. y  e.  B  ch  <->  A. w  e.  B  ps )
65ralbii 2483 . 2  |-  ( A. z  e.  A  A. y  e.  B  ch  <->  A. z  e.  A  A. w  e.  B  ps )
73, 6bitri 184 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 105   A.wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-clel 2173  df-ral 2460
This theorem is referenced by:  mhmpropd  12747
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