Theorem rspcedeq2vd 3319

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 3317 for equations, in which the right 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
rspcedeq2vd	|- ( ph -> E. x e. B C = D )

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

Allowed substitution hint:   D( x)

Proof of Theorem rspcedeq2vd

Step	Hyp	Ref	Expression
1		rspcedeqvd.1	. 2 |- ( ph -> A e. B )
2		rspcedeqvd.2	. . . 4 |- ( ( ph /\ x = A ) -> C = D )
3	2	eqcomd 2628	. . 3 |- ( ( ph /\ x = A ) -> D = C )
4	3	eqeq2d 2632	. 2 |- ( ( ph /\ x = A ) -> ( C = D <-> C = C ) )
5		eqidd 2623	. 2 |- ( ph -> C = C )
6	1, 4, 5	rspcedvd 3317	1 |- ( ph -> E. x e. B C = D )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202

This theorem is referenced by:  symgextfo  17842  smatvscl  20330  eucrctshift  27103  ntrclsneine0lem  38362  mogoldbblem  41629  sbgoldbwt  41665  sbgoldbo  41675