Theorem ovanraleqv 5920

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

Step	Hyp	Ref	Expression
1		ovanraleqv.1	. . 3 |- ( B = X -> ( ph <-> ps ) )
2		oveq2 5904	. . . 4 |- ( B = X -> ( A .x. B ) = ( A .x. X ) )
3	2	eqeq1d 2198	. . 3 |- ( B = X -> ( ( A .x. B ) = C <-> ( A .x. X ) = C ) )
4	1, 3	anbi12d 473	. 2 |- ( B = X -> ( ( ph /\ ( A .x. B ) = C ) <-> ( ps /\ ( A .x. X ) = C ) ) )
5	4	ralbidv 2490	1 |- ( B = X -> ( A. x e. V ( ph /\ ( A .x. B ) = C ) <-> A. x e. V ( ps /\ ( A .x. X ) = C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   A.wral 2468  (class class class)co 5896

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5899

This theorem is referenced by:  mgmidmo  12848  ismgmid  12853  ismgmid2  12856  mgmidsssn0  12860  sgrpidmndm  12881  ismndd  12898  mnd1  12907  mhmmnd  13058