Theorem rspcedeq1vd 2793

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2790 for equations, in which the left hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.)

Hypotheses

Ref	Expression
rspcedeqvd.1	|- ( ph -> A e. B )
rspcedeqvd.2	|- ( ( ph /\ x = A ) -> C = D )

Assertion

Ref	Expression
rspcedeq1vd	|- ( ph -> E. x e. B C = D )

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

Allowed substitution hint:   C( x)

Proof of Theorem rspcedeq1vd

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	. 2 |- ( ph -> A e. B )
2		rspcedeqvd.2	. . 3 |- ( ( ph /\ x = A ) -> C = D )
3	2	eqeq1d 2146	. 2 |- ( ( ph /\ x = A ) -> ( C = D <-> D = D ) )
4		eqidd 2138	. 2 |- ( ph -> D = D )
5	1, 3, 4	rspcedvd 2790	1 |- ( ph -> E. x e. B C = D )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   E.wrex 2415

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683

This theorem is referenced by: (None)