Theorem 19.43 1651

Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)

Assertion

Ref	Expression
19.43	|- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )

Proof of Theorem 19.43

Step	Hyp	Ref	Expression
1		hbe1 1518	. . . 4 |- ( E. x ph -> A. x E. x ph )
2		hbe1 1518	. . . 4 |- ( E. x ps -> A. x E. x ps )
3	1, 2	hbor 1569	. . 3 |- ( ( E. x ph \/ E. x ps ) -> A. x ( E. x ph \/ E. x ps ) )
4		19.8a 1613	. . . 4 |- ( ph -> E. x ph )
5		19.8a 1613	. . . 4 |- ( ps -> E. x ps )
6	4, 5	orim12i 761	. . 3 |- ( ( ph \/ ps ) -> ( E. x ph \/ E. x ps ) )
7	3, 6	exlimih 1616	. 2 |- ( E. x ( ph \/ ps ) -> ( E. x ph \/ E. x ps ) )
8		orc 714	. . . 4 |- ( ph -> ( ph \/ ps ) )
9	8	eximi 1623	. . 3 |- ( E. x ph -> E. x ( ph \/ ps ) )
10		olc 713	. . . 4 |- ( ps -> ( ph \/ ps ) )
11	10	eximi 1623	. . 3 |- ( E. x ps -> E. x ( ph \/ ps ) )
12	9, 11	jaoi 718	. 2 |- ( ( E. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) )
13	7, 12	impbii 126	1 |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )

Syntax hints:   <-> wb 105   \/ wo 710   E.wex 1515

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.44  1705  19.45  1706  19.34  1707  sborv  1914  r19.43  2664  rexun  3353  unipr  3864  uniun  3869  unopab  4123  dmun  4885  coundi  5184  coundir  5185