Theorem 19.41 1879

Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
19.41.1	|- F/xps

Assertion

Ref	Expression
19.41	|- (E.x(ph /\ ps) <-> (E.xph /\ ps))

Proof of Theorem 19.41

Step	Hyp	Ref	Expression
1		19.40 1609	. . 3 |- (E.x(ph /\ ps) -> (E.xph /\ E.xps))
2		19.41.1	. . . . 5 |- F/xps
3		id 19	. . . . 5 |- (ps -> ps)
4	2, 3	exlimi 1803	. . . 4 |- (E.xps -> ps)
5	4	anim2i 552	. . 3 |- ((E.xph /\ E.xps) -> (E.xph /\ ps))
6	1, 5	syl 15	. 2 |- (E.x(ph /\ ps) -> (E.xph /\ ps))
7		pm3.21 435	. . . 4 |- (ps -> (ph -> (ph /\ ps)))
8	2, 7	eximd 1770	. . 3 |- (ps -> (E.xph -> E.x(ph /\ ps)))
9	8	impcom 419	. 2 |- ((E.xph /\ ps) -> E.x(ph /\ ps))
10	6, 9	impbii 180	1 |- (E.x(ph /\ ps) <-> (E.xph /\ ps))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.42  1880  19.41v  1901  eean  1912  r19.41  2764  eliunxp  4822