Theorem r19.43 2666

Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)

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		df-rex 2492	. . . 4 |- ( E. x e. A ( ph \/ ps ) <-> E. x ( x e. A /\ ( ph \/ ps ) ) )
2		andi 820	. . . . 5 |- ( ( x e. A /\ ( ph \/ ps ) ) <-> ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
3	2	exbii 1629	. . . 4 |- ( E. x ( x e. A /\ ( ph \/ ps ) ) <-> E. x ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
4	1, 3	bitri 184	. . 3 |- ( E. x e. A ( ph \/ ps ) <-> E. x ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
5		19.43 1652	. . 3 |- ( E. x ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) <-> ( E. x ( x e. A /\ ph ) \/ E. x ( x e. A /\ ps ) ) )
6	4, 5	bitri 184	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x ( x e. A /\ ph ) \/ E. x ( x e. A /\ ps ) ) )
7		df-rex 2492	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2492	. . 3 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
9	7, 8	orbi12i 766	. 2 |- ( ( E. x e. A ph \/ E. x e. A ps ) <-> ( E. x ( x e. A /\ ph ) \/ E. x ( x e. A /\ ps ) ) )
10	6, 9	bitr4i 187	1 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   \/ wo 710   E.wex 1516   e. wcel 2178   E.wrex 2487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-rex 2492

This theorem is referenced by:  r19.44av  2667  r19.45av  2668  r19.45mv  3562  r19.44mv  3563  iunun  4020  ltexprlemloc  7755  pythagtriplem2  12704  pythagtrip  12721