Theorem 19.30 1604

Description: Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
19.30	|- (A.x(ph \/ ps) -> (A.xph \/ E.xps))

Proof of Theorem 19.30

Step	Hyp	Ref	Expression
1		exnal 1574	. . 3 |- (E.x -. ph <-> -. A.xph)
2		exim 1575	. . 3 |- (A.x(-. ph -> ps) -> (E.x -. ph -> E.xps))
3	1, 2	syl5bir 209	. 2 |- (A.x(-. ph -> ps) -> (-. A.xph -> E.xps))
4		df-or 359	. . 3 |- ((ph \/ ps) <-> (-. ph -> ps))
5	4	albii 1566	. 2 |- (A.x(ph \/ ps) <-> A.x(-. ph -> ps))
6		df-or 359	. 2 |- ((A.xph \/ E.xps) <-> (-. A.xph -> E.xps))
7	3, 5, 6	3imtr4i 257	1 |- (A.x(ph \/ ps) -> (A.xph \/ E.xps))

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357  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-or 359  df-ex 1542

This theorem is referenced by:  19.33b  1608