Theorem exan 1788

Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof shortened by Wolf Lammen, 8-Oct-2021.)

Hypothesis

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

Assertion

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

Proof of Theorem exan

Step	Hyp	Ref	Expression
1		exan.1	. . 3 |- ( E. x ph /\ ps )
2	1	simpli 474	. 2 |- E. x ph
3	1	simpri 478	. . . 4 |- ps
4		pm3.21 464	. . . 4 |- ( ps -> ( ph -> ( ph /\ ps ) ) )
5	3, 4	ax-mp 5	. . 3 |- ( ph -> ( ph /\ ps ) )
6	5	eximi 1762	. 2 |- ( E. x ph -> E. x ( ph /\ ps ) )
7	2, 6	ax-mp 5	1 |- E. x ( ph /\ ps )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  bm1.3ii  4784  ac6s6f  33981  fnchoice  39188