Theorem r19.43 2628

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 2454	. . . 4 |- ( E. x e. A ( ph \/ ps ) <-> E. x ( x e. A /\ ( ph \/ ps ) ) )
2		andi 813	. . . . 5 |- ( ( x e. A /\ ( ph \/ ps ) ) <-> ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
3	2	exbii 1598	. . . 4 |- ( E. x ( x e. A /\ ( ph \/ ps ) ) <-> E. x ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
4	1, 3	bitri 183	. . 3 |- ( E. x e. A ( ph \/ ps ) <-> E. x ( ( x e. A /\ ph ) \/ ( x e. A /\ ps ) ) )
5		19.43 1621	. . 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 183	. 2 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x ( x e. A /\ ph ) \/ E. x ( x e. A /\ ps ) ) )
7		df-rex 2454	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8		df-rex 2454	. . 3 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
9	7, 8	orbi12i 759	. 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 186	1 |- ( E. x e. A ( ph \/ ps ) <-> ( E. x e. A ph \/ E. x e. A ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   \/ wo 703   E.wex 1485   e. wcel 2141   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-rex 2454

This theorem is referenced by:  r19.44av  2629  r19.45av  2630  r19.45mv  3508  r19.44mv  3509  iunun  3951  ltexprlemloc  7569  pythagtriplem2  12220  pythagtrip  12237