Theorem 19.43 1628

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 1495	. . . 4 |- ( E. x ph -> A. x E. x ph )
2		hbe1 1495	. . . 4 |- ( E. x ps -> A. x E. x ps )
3	1, 2	hbor 1546	. . 3 |- ( ( E. x ph \/ E. x ps ) -> A. x ( E. x ph \/ E. x ps ) )
4		19.8a 1590	. . . 4 |- ( ph -> E. x ph )
5		19.8a 1590	. . . 4 |- ( ps -> E. x ps )
6	4, 5	orim12i 759	. . 3 |- ( ( ph \/ ps ) -> ( E. x ph \/ E. x ps ) )
7	3, 6	exlimih 1593	. 2 |- ( E. x ( ph \/ ps ) -> ( E. x ph \/ E. x ps ) )
8		orc 712	. . . 4 |- ( ph -> ( ph \/ ps ) )
9	8	eximi 1600	. . 3 |- ( E. x ph -> E. x ( ph \/ ps ) )
10		olc 711	. . . 4 |- ( ps -> ( ph \/ ps ) )
11	10	eximi 1600	. . 3 |- ( E. x ps -> E. x ( ph \/ ps ) )
12	9, 11	jaoi 716	. 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 708   E.wex 1492

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 709  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.44  1682  19.45  1683  19.34  1684  sborv  1890  r19.43  2635  rexun  3317  unipr  3825  uniun  3830  unopab  4084  dmun  4836  coundi  5132  coundir  5133