Theorem ovanraleqv 5991

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 5975	. . . 4 |- ( B = X -> ( A .x. B ) = ( A .x. X ) )
3	2	eqeq1d 2216	. . 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 2508	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 1373   A.wral 2486  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  mgmidmo  13319  ismgmid  13324  ismgmid2  13327  mgmidsssn0  13331  gsumress  13342  sgrpidmndm  13367  ismndd  13384  mnd1  13402  gsumvallem2  13440  mhmmnd  13567