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Theorem ovanraleqv 5802
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 5786 . . . 4  |-  ( B  =  X  ->  ( A  .x.  B )  =  ( A  .x.  X
) )
32eqeq1d 2149 . . 3  |-  ( B  =  X  ->  (
( A  .x.  B
)  =  C  <->  ( A  .x.  X )  =  C ) )
41, 3anbi12d 465 . 2  |-  ( B  =  X  ->  (
( ph  /\  ( A  .x.  B )  =  C )  <->  ( ps  /\  ( A  .x.  X
)  =  C ) ) )
54ralbidv 2438 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 1332   A.wral 2417  (class class class)co 5778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-un 3076  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-iota 5092  df-fv 5135  df-ov 5781
This theorem is referenced by: (None)
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