Theorem rspceeqv 2928

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 2243	. 2 |- ( x = A -> ( E = C <-> E = D ) )
3	2	rspcev 2910	1 |- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   E.wrex 2511

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804

This theorem is referenced by:  elixpsn  6903  ixpsnf1o  6904  elfir  7171  0ct  7305  ctmlemr  7306  ctssdclemn0  7308  fodju0  7345  ccats1pfxeqrex  11295  mertenslemi1  12095  mertenslem2  12096  nninfctlemfo  12610  pcprmpw  12906  1arithlem4  12938  ctiunctlemfo  13059  elrestr  13329  lss1d  14396  lspsn  14429  znf1o  14664  restopnb  14904  mopnex  15228  metrest  15229  mpodvdsmulf1o  15713  lgsquadlem1  15805  2sqlem2  15843  mul2sq  15844  2sqlem3  15845  2sqlem9  15852  2sqlem10  15853  nnnninfex  16624