Theorem dfex2 198

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

Hypothesis

Ref	Expression
dfex2.1	⊢ F:(α → ∗)

Assertion

Ref	Expression
dfex2	⊢ ⊤⊧[(∃F) = (F(εF))]

Proof of Theorem dfex2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfex2.1	. . 3 ⊢ F:(α → ∗)
2		wv 64	. . . . 5 ⊢ x:α:α
3	1, 2	ac 197	. . . 4 ⊢ (Fx:α)⊧(F(εF))
4		wtru 43	. . . 4 ⊢ ⊤:∗
5	3, 4	adantl 56	. . 3 ⊢ (⊤, (Fx:α))⊧(F(εF))
6	1, 5	exlimdv2 166	. 2 ⊢ (⊤, (∃F))⊧(F(εF))
7		wat 193	. . . . 5 ⊢ ε:((α → ∗) → α)
8	7, 1	wc 50	. . . 4 ⊢ (εF):α
9	1, 8	ax4e 168	. . 3 ⊢ (F(εF))⊧(∃F)
10	9, 4	adantl 56	. 2 ⊢ (⊤, (F(εF)))⊧(∃F)
11	6, 10	ded 84	1 ⊢ ⊤⊧[(∃F) = (F(εF))]

Syntax hints:  tv 1   → ht 2  ∗hb 3  kc 5   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12  ∃tex 123  εtat 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)