Theorem rexlimddv 2743

Description: Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)

Hypotheses

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

Assertion

Ref	Expression
rexlimddv	|- (ph -> ch)

Distinct variable groups:   ph,x   ch,x

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

Proof of Theorem rexlimddv

Step	Hyp	Ref	Expression
1		rexlimddv.1	. 2 |- (ph -> E.x e. A ps)
2		rexlimddv.2	. . 3 |- ((ph /\ (x e. A /\ ps)) -> ch)
3	2	rexlimdvaa 2740	. 2 |- (ph -> (E.x e. A ps -> ch))
4	1, 3	mpd 14	1 |- (ph -> ch)

This proof depends on syntax axioms:   -> wi 4   /\ wa 358   e. wcel 1710  E.wrex 2616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is used by: (None)