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  6890  ixpsnf1o  6891  elfir  7151  0ct  7285  ctmlemr  7286  ctssdclemn0  7288  fodju0  7325  ccats1pfxeqrex  11263  mertenslemi1  12062  mertenslem2  12063  nninfctlemfo  12577  pcprmpw  12873  1arithlem4  12905  ctiunctlemfo  13026  elrestr  13296  lss1d  14363  lspsn  14396  znf1o  14631  restopnb  14871  mopnex  15195  metrest  15196  mpodvdsmulf1o  15680  lgsquadlem1  15772  2sqlem2  15810  mul2sq  15811  2sqlem3  15812  2sqlem9  15819  2sqlem10  15820  nnnninfex  16476