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Theorem cbvaldvaw 1942
Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. Version of cbvaldva 1940 with a disjoint variable condition. (Contributed by David Moews, 1-May-2017.) (Revised by GG, 10-Jan-2024.) (Revised by Wolf Lammen, 10-Feb-2024.)
Hypothesis
Ref Expression
cbvaldvaw.1  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
cbvaldvaw  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Distinct variable groups:    ps, y    ch, x    ph, x, y
Allowed substitution hints:    ps( x)    ch( y)

Proof of Theorem cbvaldvaw
StepHypRef Expression
1 cbvaldvaw.1 . . . . . 6  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
21ancoms 268 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  ( ps  <->  ch ) )
32pm5.74da 443 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  ps )  <->  (
ph  ->  ch ) ) )
43cbvalvw 1931 . . 3  |-  ( A. x ( ph  ->  ps )  <->  A. y ( ph  ->  ch ) )
5 19.21v 1884 . . 3  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
6 19.21v 1884 . . 3  |-  ( A. y ( ph  ->  ch )  <->  ( ph  ->  A. y ch ) )
74, 5, 63bitr3i 210 . 2  |-  ( (
ph  ->  A. x ps )  <->  (
ph  ->  A. y ch )
)
87pm5.74ri 181 1  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472
This theorem is referenced by:  cbval2vw  1944
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