Theorem rspceeqv 2925

Description: Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.)

Hypothesis

Ref	Expression
rspceeqv.1	|- ( x = A -> C = D )

Assertion

Ref	Expression
rspceeqv	|- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Distinct variable groups:   x, A   x, B   x, D   x, E

Allowed substitution hint:   C( x)

Proof of Theorem rspceeqv

Step	Hyp	Ref	Expression
1		rspceeqv.1	. . 3 |- ( x = A -> C = D )
2	1	eqeq2d 2241	. 2 |- ( x = A -> ( E = C <-> E = D ) )
3	2	rspcev 2907	1 |- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801

This theorem is referenced by:  elixpsn  6895  ixpsnf1o  6896  elfir  7156  0ct  7290  ctmlemr  7291  ctssdclemn0  7293  fodju0  7330  ccats1pfxeqrex  11268  mertenslemi1  12067  mertenslem2  12068  nninfctlemfo  12582  pcprmpw  12878  1arithlem4  12910  ctiunctlemfo  13031  elrestr  13301  lss1d  14368  lspsn  14401  znf1o  14636  restopnb  14876  mopnex  15200  metrest  15201  mpodvdsmulf1o  15685  lgsquadlem1  15777  2sqlem2  15815  mul2sq  15816  2sqlem3  15817  2sqlem9  15824  2sqlem10  15825  nnnninfex  16502