Theorem 19.43 1559

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 1424	. . . 4 |- ( E. x ph -> A. x E. x ph )
2		hbe1 1424	. . . 4 |- ( E. x ps -> A. x E. x ps )
3	1, 2	hbor 1478	. . 3 |- ( ( E. x ph \/ E. x ps ) -> A. x ( E. x ph \/ E. x ps ) )
4		19.8a 1522	. . . 4 |- ( ph -> E. x ph )
5		19.8a 1522	. . . 4 |- ( ps -> E. x ps )
6	4, 5	orim12i 708	. . 3 |- ( ( ph \/ ps ) -> ( E. x ph \/ E. x ps ) )
7	3, 6	exlimih 1524	. 2 |- ( E. x ( ph \/ ps ) -> ( E. x ph \/ E. x ps ) )
8		orc 665	. . . 4 |- ( ph -> ( ph \/ ps ) )
9	8	eximi 1531	. . 3 |- ( E. x ph -> E. x ( ph \/ ps ) )
10		olc 664	. . . 4 |- ( ps -> ( ph \/ ps ) )
11	10	eximi 1531	. . 3 |- ( E. x ps -> E. x ( ph \/ ps ) )
12	9, 11	jaoi 668	. 2 |- ( ( E. x ph \/ E. x ps ) -> E. x ( ph \/ ps ) )
13	7, 12	impbii 124	1 |- ( E. x ( ph \/ ps ) <-> ( E. x ph \/ E. x ps ) )

Syntax hints:   <-> wb 103   \/ wo 661   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  19.44  1612  19.45  1613  19.34  1614  sborv  1811  r19.43  2512  rexun  3152  unipr  3615  uniun  3620  unopab  3857  dmun  4560  coundi  4842  coundir  4843