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Theorem ax12 202
Description: Axiom of Quantifier Introduction. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint).
Assertion
Ref Expression
ax12 ⊤⊧[(¬ (λz:α [z:α = x:α])) ⇒ [(¬ (λz:α [z:α = y:α])) ⇒ [[x:α = y:α] ⇒ (λz:α [x:α = y:α])]]]
Distinct variable groups:   x,z   y,z   α,z

Proof of Theorem ax12
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wv 58 . . . . . . 7 x:α:α
2 wv 58 . . . . . . 7 y:α:α
31, 2weqi 68 . . . . . 6 [x:α = y:α]:∗
4 wv 58 . . . . . . 7 p:α:α
53, 4ax-17 95 . . . . . 6 ⊤⊧[(λz:α [x:α = y:α]p:α) = [x:α = y:α]]
63, 5isfree 176 . . . . 5 ⊤⊧[[x:α = y:α] ⇒ (λz:α [x:α = y:α])]
7 wnot 128 . . . . . 6 ¬ :(∗ → ∗)
8 wal 124 . . . . . . 7 :((α → ∗) → ∗)
9 wv 58 . . . . . . . . 9 z:α:α
109, 2weqi 68 . . . . . . . 8 [z:α = y:α]:∗
1110wl 59 . . . . . . 7 λz:α [z:α = y:α]:(α → ∗)
128, 11wc 45 . . . . . 6 (λz:α [z:α = y:α]):∗
137, 12wc 45 . . . . 5 (¬ (λz:α [z:α = y:α])):∗
146, 13adantr 50 . . . 4 (⊤, (¬ (λz:α [z:α = y:α])))⊧[[x:α = y:α] ⇒ (λz:α [x:α = y:α])]
1514ex 148 . . 3 ⊤⊧[(¬ (λz:α [z:α = y:α])) ⇒ [[x:α = y:α] ⇒ (λz:α [x:α = y:α])]]
169, 1weqi 68 . . . . . 6 [z:α = x:α]:∗
1716wl 59 . . . . 5 λz:α [z:α = x:α]:(α → ∗)
188, 17wc 45 . . . 4 (λz:α [z:α = x:α]):∗
197, 18wc 45 . . 3 (¬ (λz:α [z:α = x:α])):∗
2015, 19adantr 50 . 2 (⊤, (¬ (λz:α [z:α = x:α])))⊧[(¬ (λz:α [z:α = y:α])) ⇒ [[x:α = y:α] ⇒ (λz:α [x:α = y:α])]]
2120ex 148 1 ⊤⊧[(¬ (λz:α [z:α = x:α])) ⇒ [(¬ (λz:α [z:α = y:α])) ⇒ [[x:α = y:α] ⇒ (λz:α [x:α = y:α])]]]
Colors of variables: type var term
Syntax hints:  tv 1  ht 2  hb 3  kc 5  λkl 6   = ke 7  kt 8  [kbr 9  wffMMJ2 11  ¬ tne 110  tim 111  tal 112
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-eta 165
This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120
This theorem is referenced by: (None)
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