Theorem ax4e 168

Description: Existential introduction. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypotheses

Ref	Expression
ax4e.1	|- F:(al -> *)
ax4e.2	|- A:al

Assertion

Ref	Expression
ax4e	|- (FA) |= (E.F)

Proof of Theorem ax4e

Dummy variables x p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wv 64	. . . . 5 |- p:*:*
2		ax4e.1	. . . . . . 7 |- F:(al -> *)
3		ax4e.2	. . . . . . 7 |- A:al
4	2, 3	wc 50	. . . . . 6 |- (FA):*
5		wal 134	. . . . . . 7 |- A.:((al -> *) -> *)
6		wim 137	. . . . . . . . 9 |- ==> :(* -> (* -> *))
7		wv 64	. . . . . . . . . 10 |- x:al:al
8	2, 7	wc 50	. . . . . . . . 9 |- (Fx:al):*
9	6, 8, 1	wov 72	. . . . . . . 8 |- [(Fx:al) ==> p:*]:*
10	9	wl 66	. . . . . . 7 |- \x:al [(Fx:al) ==> p:*]:(al -> *)
11	5, 10	wc 50	. . . . . 6 |- (A.\x:al [(Fx:al) ==> p:*]):*
12	4, 11	simpl 22	. . . . 5 |- ((FA), (A.\x:al [(Fx:al) ==> p:*])) |= (FA)
13	7, 3	weqi 76	. . . . . . . . . 10 |- [x:al = A]:*
14	13	id 25	. . . . . . . . 9 |- [x:al = A] |= [x:al = A]
15	2, 7, 14	ceq2 90	. . . . . . . 8 |- [x:al = A] |= [(Fx:al) = (FA)]
16	6, 8, 1, 15	oveq1 99	. . . . . . 7 |- [x:al = A] |= [[(Fx:al) ==> p:*] = [(FA) ==> p:*]]
17	9, 3, 16	cla4v 152	. . . . . 6 |- (A.\x:al [(Fx:al) ==> p:*]) |= [(FA) ==> p:*]
18	17, 4	adantl 56	. . . . 5 |- ((FA), (A.\x:al [(Fx:al) ==> p:*])) |= [(FA) ==> p:*]
19	1, 12, 18	mpd 156	. . . 4 |- ((FA), (A.\x:al [(Fx:al) ==> p:*])) |= p:*
20	19	ex 158	. . 3 |- (FA) |= [(A.\x:al [(Fx:al) ==> p:*]) ==> p:*]
21	20	alrimiv 151	. 2 |- (FA) |= (A.\p:* [(A.\x:al [(Fx:al) ==> p:*]) ==> p:*])
22	2	exval 143	. . 3 |- T. |= [(E.F) = (A.\p:* [(A.\x:al [(Fx:al) ==> p:*]) ==> p:*])]
23	4, 22	a1i 28	. 2 |- (FA) |= [(E.F) = (A.\p:* [(A.\x:al [(Fx:al) ==> p:*]) ==> p:*])]
24	21, 23	mpbir 87	1 |- (FA) |= (E.F)

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 121  A.tal 122  E.tex 123

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73  df-al 126  df-an 128  df-im 129  df-ex 131

This theorem is referenced by:  cla4ev  169  19.8a  170  dfex2  198  axrep  220