Theorem rspcedeq1vd 2886

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2883 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 2214	. 2 |- ( ( ph /\ x = A ) -> ( C = D <-> D = D ) )
4		eqidd 2206	. 2 |- ( ph -> D = D )
5	1, 3, 4	rspcedvd 2883	1 |- ( ph -> E. x e. B C = D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   E.wrex 2485

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774

This theorem is referenced by: (None)