Theorem exbi 1581

Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
exbi	|- (A.x(ph <-> ps) -> (E.xph <-> E.xps))

Proof of Theorem exbi

Step	Hyp	Ref	Expression
1		bi1 178	. . . 4 |- ((ph <-> ps) -> (ph -> ps))
2	1	alimi 1559	. . 3 |- (A.x(ph <-> ps) -> A.x(ph -> ps))
3		exim 1575	. . 3 |- (A.x(ph -> ps) -> (E.xph -> E.xps))
4	2, 3	syl 15	. 2 |- (A.x(ph <-> ps) -> (E.xph -> E.xps))
5		bi2 189	. . . 4 |- ((ph <-> ps) -> (ps -> ph))
6	5	alimi 1559	. . 3 |- (A.x(ph <-> ps) -> A.x(ps -> ph))
7		exim 1575	. . 3 |- (A.x(ps -> ph) -> (E.xps -> E.xph))
8	6, 7	syl 15	. 2 |- (A.x(ph <-> ps) -> (E.xps -> E.xph))
9	4, 8	impbid 183	1 |- (A.x(ph <-> ps) -> (E.xph <-> E.xps))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  exbii  1582  exbidh  1591  exintrbi  1613  19.19  1862