Theorem rspceeqv 2807

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 2151	. 2 |- ( x = A -> ( E = C <-> E = D ) )
3	2	rspcev 2789	1 |- ( ( A e. B /\ E = D ) -> E. x e. B E = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   E.wrex 2417

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688

This theorem is referenced by:  elixpsn  6629  ixpsnf1o  6630  elfir  6861  0ct  6992  ctmlemr  6993  ctssdclemn0  6995  fodju0  7019  mertenslemi1  11304  mertenslem2  11305  ctiunctlemfo  11952  elrestr  12128  restopnb  12350  mopnex  12674  metrest  12675