Theorem exists2 1435

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 1434	. . 3 |- (E!x x = x <-> A.x x = y)
2		pm3.24 655	. . . 4 |- -. (ph /\ -. ph)
3		ax-16 1194	. . . . . . 7 |- (A.x x = y -> (ph -> A.xph))
4	3	a5i 965	. . . . . 6 |- (A.x x = y -> A.x(ph -> A.xph))
5		19.9t 1011	. . . . . 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 1194	. . . . . . 7 |- (A.x x = y -> (-. ph -> A.x -. ph))
8	7	a5i 965	. . . . . 6 |- (A.x x = y -> A.x(-. ph -> A.x -. ph))
9		19.9t 1011	. . . . . 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 556	. . . 4 |- (A.x x = y -> ((E.xph /\ E.x -. ph) -> (ph /\ -. ph)))
12	2, 11	mtoi 107	. . 3 |- (A.x x = y -> -. (E.xph /\ E.x -. ph))
13	1, 12	sylbi 199	. 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 223  A.wal 950  E.wex 956   = wceq 1099  E!weu 1357

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-16 1194

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-eu 1359

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