Theorem rexsngf 39220

Description: Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
rexsngf.1	|- F/ x ps
rexsngf.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rexsngf	|- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   V( x)

Proof of Theorem rexsngf

Step	Hyp	Ref	Expression
1		rexsns 4217	. 2 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
2		rexsngf.1	. . 3 |- F/ x ps
3		rexsngf.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	2, 3	sbciegf 3467	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
5	1, 4	syl5bb 272	1 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   e. wcel 1990   E.wrex 2913   [.wsbc 3435   {csn 4177

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-3an 1039  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  df-sbc 3436  df-sn 4178

This theorem is referenced by:  iunxsngf2  39230