Theorem rspcebdv 3314

Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.)

Hypotheses

Ref	Expression
rspcdv.1	|- ( ph -> A e. B )
rspcdv.2	|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
rspcebdv.1	|- ( ( ph /\ ps ) -> x = A )

Assertion

Ref	Expression
rspcebdv	|- ( ph -> ( E. x e. B ps <-> ch ) )

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

Allowed substitution hint:   ps( x)

Proof of Theorem rspcebdv

Step	Hyp	Ref	Expression
1		rspcebdv.1	. . . . . . 7 |- ( ( ph /\ ps ) -> x = A )
2		rspcdv.2	. . . . . . 7 |- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
3	1, 2	syldan 487	. . . . . 6 |- ( ( ph /\ ps ) -> ( ps <-> ch ) )
4	3	biimpd 219	. . . . 5 |- ( ( ph /\ ps ) -> ( ps -> ch ) )
5	4	expcom 451	. . . 4 |- ( ps -> ( ph -> ( ps -> ch ) ) )
6	5	pm2.43b 55	. . 3 |- ( ph -> ( ps -> ch ) )
7	6	rexlimdvw 3034	. 2 |- ( ph -> ( E. x e. B ps -> ch ) )
8		rspcdv.1	. . 3 |- ( ph -> A e. B )
9	8, 2	rspcedv 3313	. 2 |- ( ph -> ( ch -> E. x e. B ps ) )
10	7, 9	impbid 202	1 |- ( ph -> ( E. x e. B ps <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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:  fusgr2wsp2nb  27198