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Theorem ovanraleqv 5893
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 5877 . . . 4  |-  ( B  =  X  ->  ( A  .x.  B )  =  ( A  .x.  X
) )
32eqeq1d 2186 . . 3  |-  ( B  =  X  ->  (
( A  .x.  B
)  =  C  <->  ( A  .x.  X )  =  C ) )
41, 3anbi12d 473 . 2  |-  ( B  =  X  ->  (
( ph  /\  ( A  .x.  B )  =  C )  <->  ( ps  /\  ( A  .x.  X
)  =  C ) ) )
54ralbidv 2477 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 104    <-> wb 105    = wceq 1353   A.wral 2455  (class class class)co 5869
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872
This theorem is referenced by:  mgmidmo  12680  ismgmid  12685  ismgmid2  12688  mgmidsssn0  12692  sgrpidmndm  12710  ismndd  12727  mnd1  12734  mhmmnd  12866
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