Theorem rexm 3550

Description: Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)

Assertion

Ref	Expression
rexm	|- ( E. x e. A ph -> E. x x e. A )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rexm

Step	Hyp	Ref	Expression
1		df-rex 2481	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpl 109	. . 3 |- ( ( x e. A /\ ph ) -> x e. A )
3	2	eximi 1614	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x x e. A )
4	1, 3	sylbi 121	1 |- ( E. x e. A ph -> E. x x e. A )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1506   e. wcel 2167   E.wrex 2476

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-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-rex 2481

This theorem is referenced by:  elrelimasn  5035  eusvobj2  5908  exmidomni  7208  fodjum  7212  ismgmid  13020  ismnd  13060  dfgrp2e  13160  zrhval  14173