Theorem rspcef 39241

Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
rspcef.1	|- F/ x ps
rspcef.2	|- F/_ x A
rspcef.3	|- F/_ x B
rspcef.4	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rspcef	|- ( ( A e. B /\ ps ) -> E. x e. B ph )

Proof of Theorem rspcef

Step	Hyp	Ref	Expression
1		rspcef.1	. 2 |- F/ x ps
2		rspcef.2	. 2 |- F/_ x A
3		rspcef.3	. 2 |- F/_ x B
4		rspcef.4	. 2 |- ( x = A -> ( ph <-> ps ) )
5	1, 2, 3, 4	rspcegf 39182	1 |- ( ( A e. B /\ ps ) -> E. x e. B ph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   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-rex 2918  df-v 3202

This theorem is referenced by:  iinssdf  39328  opnvonmbllem1  40846  smfresal  40995  smfmullem2  40999