Theorem exists2 1499

Description: A condition implying that at least two things exist.

Assertion

Ref	Expression
exists2	|- ((E.xph /\ E.x -. ph) -> -. E!x x = x)

Proof of Theorem exists2

Step	Hyp	Ref	Expression
1		exists1 1498	. . 3 |- (E!x x = x <-> A.x x = y)
2		pm3.24 661	. . . 4 |- -. (ph /\ -. ph)
3		ax-16 1247	. . . . . . 7 |- (A.x x = y -> (ph -> A.xph))
4	3	a5i 1025	. . . . . 6 |- (A.x x = y -> A.x(ph -> A.xph))
5		19.9t 1071	. . . . . 6 |- (A.x(ph -> A.xph) -> (E.xph -> ph))
6	4, 5	syl 10	. . . . 5 |- (A.x x = y -> (E.xph -> ph))
7		ax-16 1247	. . . . . . 7 |- (A.x x = y -> (-. ph -> A.x -. ph))
8	7	a5i 1025	. . . . . 6 |- (A.x x = y -> A.x(-. ph -> A.x -. ph))
9		19.9t 1071	. . . . . 6 |- (A.x(-. ph -> A.x -. ph) -> (E.x -. ph -> -. ph))
10	8, 9	syl 10	. . . . 5 |- (A.x x = y -> (E.x -. ph -> -. ph))
11	6, 10	anim12d 561	. . . 4 |- (A.x x = y -> ((E.xph /\ E.x -. ph) -> (ph /\ -. ph)))
12	2, 11	mtoi 106	. . 3 |- (A.x x = y -> -. (E.xph /\ E.x -. ph))
13	1, 12	sylbi 197	. 2 |- (E!x x = x -> -. (E.xph /\ E.x -. ph))
14	13	con2i 97	1 |- ((E.xph /\ E.x -. ph) -> -. E!x x = x)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016  E!weu 1419

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-10 1002  ax-12 1004  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-eu 1421

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