Higher-Order Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HOLE Home  >  Th. List  >  dfex2 GIF version

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) = (FF))]

Proof of Theorem dfex2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 dfex2.1 . . 3 F:(α → ∗)
2 wv 64 . . . . 5 x:α:α
31, 2ac 197 . . . 4 (Fx:α)⊧(FF))
4 wtru 43 . . . 4 ⊤:∗
53, 4adantl 56 . . 3 (⊤, (Fx:α))⊧(FF))
61, 5exlimdv2 166 . 2 (⊤, (F))⊧(FF))
7 wat 193 . . . . 5 ε:((α → ∗) → α)
87, 1wc 50 . . . 4 F):α
91, 8ax4e 168 . . 3 (FF))⊧(F)
109, 4adantl 56 . 2 (⊤, (FF)))⊧(F)
116, 10ded 84 1 ⊤⊧[(F) = (FF))]
 Colors of variables: type var term 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)
 Copyright terms: Public domain W3C validator