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Theorem alval 142
 Description: Value of the for all predicate. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypothesis
Ref Expression
alval.1 F:(α → ∗)
Assertion
Ref Expression
alval ⊤⊧[(F) = [F = λx:α ⊤]]
Distinct variable group:   α,x

Proof of Theorem alval
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wal 134 . . 3 :((α → ∗) → ∗)
2 alval.1 . . 3 F:(α → ∗)
31, 2wc 50 . 2 (F):∗
4 df-al 126 . . 3 ⊤⊧[ = λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]]
51, 2, 4ceq1 89 . 2 ⊤⊧[(F) = (λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]F)]
6 wv 64 . . . 4 p:(α → ∗):(α → ∗)
7 wtru 43 . . . . 5 ⊤:∗
87wl 66 . . . 4 λx:α ⊤:(α → ∗)
96, 8weqi 76 . . 3 [p:(α → ∗) = λx:α ⊤]:∗
10 weq 41 . . . 4 = :((α → ∗) → ((α → ∗) → ∗))
116, 2weqi 76 . . . . 5 [p:(α → ∗) = F]:∗
1211id 25 . . . 4 [p:(α → ∗) = F]⊧[p:(α → ∗) = F]
1310, 6, 8, 12oveq1 99 . . 3 [p:(α → ∗) = F]⊧[[p:(α → ∗) = λx:α ⊤] = [F = λx:α ⊤]]
149, 2, 13cl 116 . 2 ⊤⊧[(λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]F) = [F = λx:α ⊤]]
153, 5, 14eqtri 95 1 ⊤⊧[(F) = [F = λx: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  wffMMJ2t 12  ∀tal 122 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-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113 This theorem depends on definitions:  df-ov 73  df-al 126 This theorem is referenced by:  ax4g  149  alrimiv  151  olc  164  orc  165  alrimi  182
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