Theorem exellim 33192

Description: Closed form of exellimddv 33193. See also exlimim 33189 for a more general theorem. (Contributed by ML, 17-Jul-2020.)

Assertion

Ref	Expression
exellim	|- ( ( E. x x e. A /\ A. x ( x e. A -> ph ) ) -> ph )

Distinct variable group:   ph, x

Allowed substitution hint:   A( x)

Proof of Theorem exellim

Step	Hyp	Ref	Expression
1		nfa1 2028	. . 3 |- F/ x A. x ( x e. A -> ph )
2		nfv 1843	. . 3 |- F/ x ph
3		sp 2053	. . 3 |- ( A. x ( x e. A -> ph ) -> ( x e. A -> ph ) )
4	1, 2, 3	exlimd 2087	. 2 |- ( A. x ( x e. A -> ph ) -> ( E. x x e. A -> ph ) )
5	4	impcom 446	1 |- ( ( E. x x e. A /\ A. x ( x e. A -> ph ) ) -> ph )

Syntax hints:   -> wi 4   /\ wa 384   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:  exellimddv  33193