Theorem rexlimddv2 40049

Description: Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
rexlimddv2.1	|- ( ph -> E. x e. A ps )
rexlimddv2.2	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )

Assertion

Ref	Expression
rexlimddv2	|- ( ph -> ch )

Distinct variable groups:   ch, x   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem rexlimddv2

Step	Hyp	Ref	Expression
1		rexlimddv2.1	. 2 |- ( ph -> E. x e. A ps )
2		rexlimddv2.2	. . 3 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
3	2	anasss 679	. 2 |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch )
4	1, 3	rexlimddv 3035	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   E.wrex 2913

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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by:  climxlim2lem  40071