Theorem exan 1301

Description: Place a conjunct in the scope of an existential quantifier. (The proof was shortened by Andrew Salmon, 25-May-2011.)

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 1201	. . . 4 |- (E.xph -> A.xE.xph)
2	1	19.28 1258	. . 3 |- (A.x(E.xph /\ ps) <-> (E.xph /\ A.xps))
3		exan.1	. . 3 |- (E.xph /\ ps)
4	2, 3	mpgbi 1171	. 2 |- (E.xph /\ A.xps)
5		19.29r 1261	. 2 |- ((E.xph /\ A.xps) -> E.x(ph /\ ps))
6	4, 5	ax-mp 7	1 |- E.x(ph /\ ps)

Syntax hints:   /\ wa 239  A.wal 1134  E.wex 1164

This theorem is referenced by:  bm1.3ii 3256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1143  ax-4 1157  ax-5o 1159  ax-6o 1162

This theorem depends on definitions:  df-bi 163  df-an 241  df-ex 1165

Copyright terms: Public domain