Theorem r19.43 2767

Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.43	|- (E.x e. A (ph \/ ps) <-> (E.x e. A ph \/ E.x e. A ps))

Proof of Theorem r19.43

Step	Hyp	Ref	Expression
1		r19.35 2759	. 2 |- (E.x e. A (-. ph -> ps) <-> (A.x e. A -. ph -> E.x e. A ps))
2		df-or 359	. . 3 |- ((ph \/ ps) <-> (-. ph -> ps))
3	2	rexbii 2640	. 2 |- (E.x e. A (ph \/ ps) <-> E.x e. A (-. ph -> ps))
4		df-or 359	. . 3 |- ((E.x e. A ph \/ E.x e. A ps) <-> (-. E.x e. A ph -> E.x e. A ps))
5		ralnex 2625	. . . 4 |- (A.x e. A -. ph <-> -. E.x e. A ph)
6	5	imbi1i 315	. . 3 |- ((A.x e. A -. ph -> E.x e. A ps) <-> (-. E.x e. A ph -> E.x e. A ps))
7	4, 6	bitr4i 243	. 2 |- ((E.x e. A ph \/ E.x e. A ps) <-> (A.x e. A -. ph -> E.x e. A ps))
8	1, 3, 7	3bitr4i 268	1 |- (E.x e. A (ph \/ ps) <-> (E.x e. A ph \/ E.x e. A ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ 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:  r19.44av  2768  r19.45av  2769  r19.45zv  3648  iunun  4047  nncdiv3  6278