Theorem exanOLD 1464

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

Hypothesis

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

Assertion

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

Proof of Theorem exanOLD

Step	Hyp	Ref	Expression
1		hbe1 1363	. . . . 5 |- (E.xph -> A.xE.xph)
2	1	19.27 1419	. . . 4 |- (A.x(ps /\ E.xph) <-> (A.xps /\ E.xph))
3		exan.1	. . . . 5 |- (E.xph /\ ps)
4		ancom 482	. . . . 5 |- ((E.xph /\ ps) <-> (ps /\ E.xph))
5	3, 4	mpbi 206	. . . 4 |- (ps /\ E.xph)
6	2, 5	mpgbi 1333	. . 3 |- (A.xps /\ E.xph)
7		19.29 1421	. . 3 |- ((A.xps /\ E.xph) -> E.x(ps /\ ph))
8	6, 7	ax-mp 7	. 2 |- E.x(ps /\ ph)
9		exancom 1401	. 2 |- (E.x(ps /\ ph) <-> E.x(ph /\ ps))
10	8, 9	mpbi 206	1 |- E.x(ph /\ ps)

Syntax hints:   /\ wa 240  A.wal 1296  E.wex 1326

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-6o 1324

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

Copyright terms: Public domain