Theorem 19.43 1565

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 1430	. . . 4 |- ( E. x ph -> A. x E. x ph )
2		hbe1 1430	. . . 4 |- ( E. x ps -> A. x E. x ps )
3	1, 2	hbor 1484	. . 3 |- ( ( E. x ph \/ E. x ps ) -> A. x ( E. x ph \/ E. x ps ) )
4		19.8a 1528	. . . 4 |- ( ph -> E. x ph )
5		19.8a 1528	. . . 4 |- ( ps -> E. x ps )
6	4, 5	orim12i 712	. . 3 |- ( ( ph \/ ps ) -> ( E. x ph \/ E. x ps ) )
7	3, 6	exlimih 1530	. 2 |- ( E. x ( ph \/ ps ) -> ( E. x ph \/ E. x ps ) )
8		orc 669	. . . 4 |- ( ph -> ( ph \/ ps ) )
9	8	eximi 1537	. . 3 |- ( E. x ph -> E. x ( ph \/ ps ) )
10		olc 668	. . . 4 |- ( ps -> ( ph \/ ps ) )
11	10	eximi 1537	. . 3 |- ( E. x ps -> E. x ( ph \/ ps ) )
12	9, 11	jaoi 672	. 2 |- ( ( E. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) )
13	7, 12	impbii 125	1 |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )

Syntax hints:   <-> wb 104   \/ wo 665   E.wex 1427

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.44  1618  19.45  1619  19.34  1620  sborv  1819  r19.43  2526  rexun  3181  unipr  3673  uniun  3678  unopab  3923  dmun  4656  coundi  4945  coundir  4946