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Theorem ovanraleqv 5730
Description: Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
Hypothesis
Ref Expression
ovanraleqv.1  |-  ( B  =  X  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ovanraleqv  |-  ( B  =  X  ->  ( A. x  e.  V  ( ph  /\  ( A 
.x.  B )  =  C )  <->  A. x  e.  V  ( ps  /\  ( A  .x.  X
)  =  C ) ) )
Distinct variable groups:    x, B    x, X
Allowed substitution hints:    ph( x)    ps( x)    A( x)    C( x)    .x. ( x)    V( x)

Proof of Theorem ovanraleqv
StepHypRef Expression
1 ovanraleqv.1 . . 3  |-  ( B  =  X  ->  ( ph 
<->  ps ) )
2 oveq2 5714 . . . 4  |-  ( B  =  X  ->  ( A  .x.  B )  =  ( A  .x.  X
) )
32eqeq1d 2108 . . 3  |-  ( B  =  X  ->  (
( A  .x.  B
)  =  C  <->  ( A  .x.  X )  =  C ) )
41, 3anbi12d 460 . 2  |-  ( B  =  X  ->  (
( ph  /\  ( A  .x.  B )  =  C )  <->  ( ps  /\  ( A  .x.  X
)  =  C ) ) )
54ralbidv 2396 1  |-  ( B  =  X  ->  ( A. x  e.  V  ( ph  /\  ( A 
.x.  B )  =  C )  <->  A. x  e.  V  ( ps  /\  ( A  .x.  X
)  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299   A.wral 2375  (class class class)co 5706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067  df-ov 5709
This theorem is referenced by: (None)
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