Theorem dfex2 185

Description: Alternative definition of the "there exists" quantifier.

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 58	. . . . 5 ⊢ x:α:α
3	1, 2	ac 184	. . . 4 ⊢ (Fx:α)⊧(F(εF))
4		wtru 40	. . . 4 ⊢ ⊤:∗
5	3, 4	adantl 51	. . 3 ⊢ (⊤, (Fx:α))⊧(F(εF))
6	1, 5	exlimdv2 156	. 2 ⊢ (⊤, (∃F))⊧(F(εF))
7		wat 180	. . . . 5 ⊢ ε:((α → ∗) → α)
8	7, 1	wc 45	. . . 4 ⊢ (εF):α
9	1, 8	ax4e 158	. . 3 ⊢ (F(εF))⊧(∃F)
10	9, 4	adantl 51	. 2 ⊢ (⊤, (F(εF)))⊧(∃F)
11	6, 10	ded 74	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 113  εtat 179

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-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-ac 183

This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119  df-ex 121

This theorem is referenced by: (None)