Theorem 19.33b 1608

Description: The antecedent provides a condition implying the converse of 19.33 1607. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)

Assertion

Ref	Expression
19.33b	|- (-. (E.xph /\ E.xps) -> (A.x(ph \/ ps) <-> (A.xph \/ A.xps)))

Proof of Theorem 19.33b

Step	Hyp	Ref	Expression
1		ianor 474	. . 3 |- (-. (E.xph /\ E.xps) <-> (-. E.xph \/ -. E.xps))
2		alnex 1543	. . . . . 6 |- (A.x -. ph <-> -. E.xph)
3		pm2.53 362	. . . . . . 7 |- ((ph \/ ps) -> (-. ph -> ps))
4	3	al2imi 1561	. . . . . 6 |- (A.x(ph \/ ps) -> (A.x -. ph -> A.xps))
5	2, 4	syl5bir 209	. . . . 5 |- (A.x(ph \/ ps) -> (-. E.xph -> A.xps))
6		olc 373	. . . . 5 |- (A.xps -> (A.xph \/ A.xps))
7	5, 6	syl6com 31	. . . 4 |- (-. E.xph -> (A.x(ph \/ ps) -> (A.xph \/ A.xps)))
8		19.30 1604	. . . . . . 7 |- (A.x(ph \/ ps) -> (A.xph \/ E.xps))
9	8	orcomd 377	. . . . . 6 |- (A.x(ph \/ ps) -> (E.xps \/ A.xph))
10	9	ord 366	. . . . 5 |- (A.x(ph \/ ps) -> (-. E.xps -> A.xph))
11		orc 374	. . . . 5 |- (A.xph -> (A.xph \/ A.xps))
12	10, 11	syl6com 31	. . . 4 |- (-. E.xps -> (A.x(ph \/ ps) -> (A.xph \/ A.xps)))
13	7, 12	jaoi 368	. . 3 |- ((-. E.xph \/ -. E.xps) -> (A.x(ph \/ ps) -> (A.xph \/ A.xps)))
14	1, 13	sylbi 187	. 2 |- (-. (E.xph /\ E.xps) -> (A.x(ph \/ ps) -> (A.xph \/ A.xps)))
15		19.33 1607	. 2 |- ((A.xph \/ A.xps) -> A.x(ph \/ ps))
16	14, 15	impbid1 194	1 |- (-. (E.xph /\ E.xps) -> (A.x(ph \/ ps) <-> (A.xph \/ A.xps)))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358  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-an 360  df-ex 1542

This theorem is referenced by: (None)