Theorem rexm 3514

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 2454	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
2		simpl 108	. . 3 |- ( ( x e. A /\ ph ) -> x e. A )
3	2	eximi 1593	. 2 |- ( E. x ( x e. A /\ ph ) -> E. x x e. A )
4	1, 3	sylbi 120	1 |- ( E. x e. A ph -> E. x x e. A )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1485   e. wcel 2141   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-rex 2454

This theorem is referenced by:  eusvobj2  5839  exmidomni  7118  fodjum  7122  ismgmid  12631  ismnd  12655  dfgrp2e  12733