Theorem ovanraleqv 5877

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 5861	. . . 4 |- ( B = X -> ( A .x. B ) = ( A .x. X ) )
3	2	eqeq1d 2179	. . 3 |- ( B = X -> ( ( A .x. B ) = C <-> ( A .x. X ) = C ) )
4	1, 3	anbi12d 470	. 2 |- ( B = X -> ( ( ph /\ ( A .x. B ) = C ) <-> ( ps /\ ( A .x. X ) = C ) ) )
5	4	ralbidv 2470	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 103   <-> wb 104   = wceq 1348   A.wral 2448  (class class class)co 5853

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  mgmidmo  12626  ismgmid  12631  ismgmid2  12634  mgmidsssn0  12638  sgrpidmndm  12656  ismndd  12673  mnd1  12679