Theorem exan 784

Description: Place a conjunct in the scope of an existential quantifier.

Hypothesis

Ref	Expression
exan.1	|- (E.xph /\ ps)

Assertion

Ref	Expression
exan	|- E.x(ph /\ ps)

Proof of Theorem exan

Step	Hyp	Ref	Expression
1		hbe1 709	. . . . 5 |- (E.xph -> A.xE.xph)
2	1	19.27 750	. . . 4 |- (A.x(ps /\ E.xph) <-> (A.xps /\ E.xph))
3		exan.1	. . . . 5 |- (E.xph /\ ps)
4		ancom 333	. . . . 5 |- ((E.xph /\ ps) <-> (ps /\ E.xph))
5	3, 4	mpbi 164	. . . 4 |- (ps /\ E.xph)
6	2, 5	mpgbi 685	. . 3 |- (A.xps /\ E.xph)
7		19.29 752	. . 3 |- ((A.xps /\ E.xph) -> E.x(ps /\ ph))
8	6, 7	ax-mp 6	. 2 |- E.x(ps /\ ph)
9		exancom 736	. 2 |- (E.x(ps /\ ph) <-> E.x(ph /\ ps))
10	8, 9	mpbi 164	1 |- E.x(ph /\ ps)

Syntax hints:   /\ wa 196  A.wal 672  E.wex 678

This theorem is referenced by:  bm1.3ii 1481

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org