Theorem r19.30 2757

Description: Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)

Assertion

Ref	Expression
r19.30	|- (A.x e. A (ph \/ ps) -> (A.x e. A ph \/ E.x e. A ps))

Proof of Theorem r19.30

Step	Hyp	Ref	Expression
1		ralim 2686	. 2 |- (A.x e. A (-. ps -> ph) -> (A.x e. A -. ps -> A.x e. A ph))
2		orcom 376	. . . 4 |- ((ph \/ ps) <-> (ps \/ ph))
3		df-or 359	. . . 4 |- ((ps \/ ph) <-> (-. ps -> ph))
4	2, 3	bitri 240	. . 3 |- ((ph \/ ps) <-> (-. ps -> ph))
5	4	ralbii 2639	. 2 |- (A.x e. A (ph \/ ps) <-> A.x e. A (-. ps -> ph))
6		orcom 376	. . 3 |- ((A.x e. A ph \/ -. A.x e. A -. ps) <-> (-. A.x e. A -. ps \/ A.x e. A ph))
7		dfrex2 2628	. . . 4 |- (E.x e. A ps <-> -. A.x e. A -. ps)
8	7	orbi2i 505	. . 3 |- ((A.x e. A ph \/ E.x e. A ps) <-> (A.x e. A ph \/ -. A.x e. A -. ps))
9		imor 401	. . 3 |- ((A.x e. A -. ps -> A.x e. A ph) <-> (-. A.x e. A -. ps \/ A.x e. A ph))
10	6, 8, 9	3bitr4i 268	. 2 |- ((A.x e. A ph \/ E.x e. A ps) <-> (A.x e. A -. ps -> A.x e. A ph))
11	1, 5, 10	3imtr4i 257	1 |- (A.x e. A (ph \/ ps) -> (A.x e. A ph \/ E.x e. A ps))

Syntax hints:  -. wn 3   -> wi 4   \/ wo 357  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by: (None)