Theorem r19.43 1304

Description: Restricted version of Theorem 19.43 of [Margaris] p. 90.

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		andi 456	. . . 4 |- ((x e. A /\ (ph \/ ps)) <-> ((x e. A /\ ph) \/ (x e. A /\ ps)))
2	1	biex 733	. . 3 |- (E.x(x e. A /\ (ph \/ ps)) <-> E.x((x e. A /\ ph) \/ (x e. A /\ ps)))
3		19.43 767	. . 3 |- (E.x((x e. A /\ ph) \/ (x e. A /\ ps)) <-> (E.x(x e. A /\ ph) \/ E.x(x e. A /\ ps)))
4	2, 3	bitr 151	. 2 |- (E.x(x e. A /\ (ph \/ ps)) <-> (E.x(x e. A /\ ph) \/ E.x(x e. A /\ ps)))
5		df-rex 1206	. 2 |- (E.x e. A (ph \/ ps) <-> E.x(x e. A /\ (ph \/ ps)))
6		df-rex 1206	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
7		df-rex 1206	. . 3 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
8	6, 7	orbi12i 216	. 2 |- ((E.x e. A ph \/ E.x e. A ps) <-> (E.x(x e. A /\ ph) \/ E.x(x e. A /\ ps)))
9	4, 5, 8	3bitr4 158	1 |- (E.x e. A (ph \/ ps) <-> (E.x e. A ph \/ E.x e. A ps))

Syntax hints:   <-> wb 127   \/ wo 195   /\ wa 196  E.wex 678   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  r19.44av 1305  r19.45av 1306  r19.45zv 1770

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  df-rex 1206

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