Theorem 19.30 764

Description: Theorem 19.30 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.30

Step	Hyp	Ref	Expression
1		19.20 690	. 2 |- (A.x(-. ps -> ph) -> (A.x -. ps -> A.xph))
2		orcom 209	. . . 4 |- ((ph \/ ps) <-> (ps \/ ph))
3		df-or 197	. . . 4 |- ((ps \/ ph) <-> (-. ps -> ph))
4	2, 3	bitr 151	. . 3 |- ((ph \/ ps) <-> (-. ps -> ph))
5	4	bial 695	. 2 |- (A.x(ph \/ ps) <-> A.x(-. ps -> ph))
6		orcom 209	. . 3 |- ((A.xph \/ -. A.x -. ps) <-> (-. A.x -. ps \/ A.xph))
7		df-ex 679	. . . 4 |- (E.xps <-> -. A.x -. ps)
8	7	orbi2i 214	. . 3 |- ((A.xph \/ E.xps) <-> (A.xph \/ -. A.x -. ps))
9		imor 204	. . 3 |- ((A.x -. ps -> A.xph) <-> (-. A.x -. ps \/ A.xph))
10	6, 8, 9	3bitr4 158	. 2 |- ((A.xph \/ E.xps) <-> (A.x -. ps -> A.xph))
11	1, 5, 10	3imtr4 192	1 |- (A.x(ph \/ ps) -> (A.xph \/ E.xps))

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195  A.wal 672  E.wex 678

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

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