Theorem bnj1109 30857

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1109.1	|- E. x ( ( A =/= B /\ ph ) -> ps )
bnj1109.2	|- ( ( A = B /\ ph ) -> ps )

Assertion

Ref	Expression
bnj1109	|- E. x ( ph -> ps )

Proof of Theorem bnj1109

Step	Hyp	Ref	Expression
1		bnj1109.2	. . . . . . 7 |- ( ( A = B /\ ph ) -> ps )
2	1	ex 450	. . . . . 6 |- ( A = B -> ( ph -> ps ) )
3	2	a1i 11	. . . . 5 |- ( ( A =/= B -> ( ph -> ps ) ) -> ( A = B -> ( ph -> ps ) ) )
4	3	ax-gen 1722	. . . 4 |- A. x ( ( A =/= B -> ( ph -> ps ) ) -> ( A = B -> ( ph -> ps ) ) )
5		bnj1109.1	. . . . 5 |- E. x ( ( A =/= B /\ ph ) -> ps )
6		impexp 462	. . . . . 6 |- ( ( ( A =/= B /\ ph ) -> ps ) <-> ( A =/= B -> ( ph -> ps ) ) )
7	6	exbii 1774	. . . . 5 |- ( E. x ( ( A =/= B /\ ph ) -> ps ) <-> E. x ( A =/= B -> ( ph -> ps ) ) )
8	5, 7	mpbi 220	. . . 4 |- E. x ( A =/= B -> ( ph -> ps ) )
9		exintr 1819	. . . 4 |- ( A. x ( ( A =/= B -> ( ph -> ps ) ) -> ( A = B -> ( ph -> ps ) ) ) -> ( E. x ( A =/= B -> ( ph -> ps ) ) -> E. x ( ( A =/= B -> ( ph -> ps ) ) /\ ( A = B -> ( ph -> ps ) ) ) ) )
10	4, 8, 9	mp2 9	. . 3 |- E. x ( ( A =/= B -> ( ph -> ps ) ) /\ ( A = B -> ( ph -> ps ) ) )
11		exancom 1787	. . 3 |- ( E. x ( ( A =/= B -> ( ph -> ps ) ) /\ ( A = B -> ( ph -> ps ) ) ) <-> E. x ( ( A = B -> ( ph -> ps ) ) /\ ( A =/= B -> ( ph -> ps ) ) ) )
12	10, 11	mpbi 220	. 2 |- E. x ( ( A = B -> ( ph -> ps ) ) /\ ( A =/= B -> ( ph -> ps ) ) )
13		df-ne 2795	. . . 4 |- ( A =/= B <-> -. A = B )
14	13	imbi1i 339	. . 3 |- ( ( A =/= B -> ( ph -> ps ) ) <-> ( -. A = B -> ( ph -> ps ) ) )
15		pm2.61 183	. . . 4 |- ( ( A = B -> ( ph -> ps ) ) -> ( ( -. A = B -> ( ph -> ps ) ) -> ( ph -> ps ) ) )
16	15	imp 445	. . 3 |- ( ( ( A = B -> ( ph -> ps ) ) /\ ( -. A = B -> ( ph -> ps ) ) ) -> ( ph -> ps ) )
17	14, 16	sylan2b 492	. 2 |- ( ( ( A = B -> ( ph -> ps ) ) /\ ( A =/= B -> ( ph -> ps ) ) ) -> ( ph -> ps ) )
18	12, 17	bnj101 30789	1 |- E. x ( ph -> ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ne 2795

This theorem is referenced by:  bnj1030  31055  bnj1128  31058  bnj1145  31061