Theorem exan 1106

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 1016	. . . . 5 |- (E.xph -> A.xE.xph)
2	1	19.27 1069	. . . 4 |- (A.x(ps /\ E.xph) <-> (A.xps /\ E.xph))
3		exan.1	. . . . 5 |- (E.xph /\ ps)
4		ancom 435	. . . . 5 |- ((E.xph /\ ps) <-> (ps /\ E.xph))
5	3, 4	mpbi 189	. . . 4 |- (ps /\ E.xph)
6	2, 5	mpgbi 987	. . 3 |- (A.xps /\ E.xph)
7		19.29 1071	. . 3 |- ((A.xps /\ E.xph) -> E.x(ps /\ ph))
8	6, 7	ax-mp 7	. 2 |- E.x(ps /\ ph)
9		exancom 1054	. 2 |- (E.x(ps /\ ph) <-> E.x(ph /\ ps))
10	8, 9	mpbi 189	1 |- E.x(ph /\ ps)

Syntax hints:   /\ wa 223  A.wal 954  E.wex 980

This theorem is referenced by:  bm1.3ii 2706

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

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