Theorem 19.30 1081

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		19.20 991	. 2 |- (A.x(-. ps -> ph) -> (A.x -. ps -> A.xph))
2		orcom 246	. . . 4 |- ((ph \/ ps) <-> (ps \/ ph))
3		df-or 224	. . . 4 |- ((ps \/ ph) <-> (-. ps -> ph))
4	2, 3	bitr 173	. . 3 |- ((ph \/ ps) <-> (-. ps -> ph))
5	4	albii 996	. 2 |- (A.x(ph \/ ps) <-> A.x(-. ps -> ph))
6		orcom 246	. . 3 |- ((A.xph \/ -. A.x -. ps) <-> (-. A.x -. ps \/ A.xph))
7		df-ex 978	. . . 4 |- (E.xps <-> -. A.x -. ps)
8	7	orbi2i 255	. . 3 |- ((A.xph \/ E.xps) <-> (A.xph \/ -. A.x -. ps))
9		imor 234	. . 3 |- ((A.x -. ps -> A.xph) <-> (-. A.x -. ps \/ A.xph))
10	6, 8, 9	3bitr4 183	. 2 |- ((A.xph \/ E.xps) <-> (A.x -. ps -> A.xph))
11	1, 5, 10	3imtr4 219	1 |- (A.x(ph \/ ps) -> (A.xph \/ E.xps))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222  A.wal 951  E.wex 977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978

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