Theorem dfex2 198

Description: Alternative definition of the "there exists" quantifier. (Contributed by Mario Carneiro, 10-Oct-2014.)

Hypothesis

Ref	Expression
dfex2.1	|- F:(al -> *)

Assertion

Ref	Expression
dfex2	|- T. |= [(E.F) = (F(@F))]

Proof of Theorem dfex2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfex2.1	. . 3 |- F:(al -> *)
2		wv 64	. . . . 5 |- x:al:al
3	1, 2	ac 197	. . . 4 |- (Fx:al) |= (F(@F))
4		wtru 43	. . . 4 |- T.:*
5	3, 4	adantl 56	. . 3 |- (T., (Fx:al)) |= (F(@F))
6	1, 5	exlimdv2 166	. 2 |- (T., (E.F)) |= (F(@F))
7		wat 193	. . . . 5 |- @:((al -> *) -> al)
8	7, 1	wc 50	. . . 4 |- (@F):al
9	1, 8	ax4e 168	. . 3 |- (F(@F)) |= (E.F)
10	9, 4	adantl 56	. 2 |- (T., (F(@F))) |= (E.F)
11	6, 10	ded 84	1 |- T. |= [(E.F) = (F(@F))]

Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  E.tex 123  @tat 191

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  ax-wat 192  ax-ac 196

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: (None)