Theorem exists2 1073

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 1072	. . 3 |- (E!x x = x <-> A.x x = y)
2		pm3.24 496	. . . 4 |- -. (ph /\ -. ph)
3		ax-16 922	. . . . . . 7 |- (A.x x = y -> (ph -> A.xph))
4	3	a5i 687	. . . . . 6 |- (A.x x = y -> A.x(ph -> A.xph))
5		19.9t 719	. . . . . 6 |- (A.x(ph -> A.xph) -> (E.xph -> ph))
6	4, 5	syl 12	. . . . 5 |- (A.x x = y -> (E.xph -> ph))
7		ax-16 922	. . . . . . 7 |- (A.x x = y -> (-. ph -> A.x -. ph))
8	7	a5i 687	. . . . . 6 |- (A.x x = y -> A.x(-. ph -> A.x -. ph))
9		19.9t 719	. . . . . 6 |- (A.x(-. ph -> A.x -. ph) -> (E.x -. ph -> -. ph))
10	8, 9	syl 12	. . . . 5 |- (A.x x = y -> (E.x -. ph -> -. ph))
11	6, 10	anim12d 431	. . . 4 |- (A.x x = y -> ((E.xph /\ E.x -. ph) -> (ph /\ -. ph)))
12	2, 11	mtoi 94	. . 3 |- (A.x x = y -> -. (E.xph /\ E.x -. ph))
13	1, 12	sylbi 174	. 2 |- (E!x x = x -> -. (E.xph /\ E.x -. ph))
14	13	con2i 89	1 |- ((E.xph /\ E.x -. ph) -> -. E!x x = x)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797  E!weu 1007

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802  ax-16 922

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-eu 1009

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