Theorem ovanraleqv 6082

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 6066	. . . 4 |- ( B = X -> ( A .x. B ) = ( A .x. X ) )
3	2	eqeq1d 2243	. . 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 2544	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 1398   A.wral 2522  (class class class)co 6058

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  mgmidmo  13635  ismgmid  13640  ismgmid2  13643  mgmidsssn0  13647  gsumress  13658  sgrpidmndm  13681  ismndd  13698  mnd1  13710  gsumvallem2  13748  mhmmnd  13869