Theorem exellimddv 33193

Description: Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 33192 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.)

Hypotheses

Ref	Expression
exellimddv.1	|- ( ph -> E. x x e. A )
exellimddv.2	|- ( ph -> ( x e. A -> ps ) )

Assertion

Ref	Expression
exellimddv	|- ( ph -> ps )

Distinct variable groups:   ph, x   ps, x

Allowed substitution hint:   A( x)

Proof of Theorem exellimddv

Step	Hyp	Ref	Expression
1		exellimddv.1	. 2 |- ( ph -> E. x x e. A )
2		exellimddv.2	. . 3 |- ( ph -> ( x e. A -> ps ) )
3	2	alrimiv 1855	. 2 |- ( ph -> A. x ( x e. A -> ps ) )
4		exellim 33192	. 2 |- ( ( E. x x e. A /\ A. x ( x e. A -> ps ) ) -> ps )
5	1, 3, 4	syl2anc 693	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704   e. wcel 1990

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-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  topdifinffinlem  33195