Theorem rexm 3627

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 2534	. 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 1653	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   E.wex 1545   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-rex 2534

This theorem is used by:  elrelimasn  5153  eusvobj2  6071  exmidomni  7482  fodjum  7486  ismgmid  13697  ismnd  13732  dfgrp2e  13833  zrhval  14952  ralsmd  17138  alsralrex  17153