Theorem alval 132

Description: Value of the for all predicate.

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.

Step	Hyp	Ref	Expression
1		wal 124	. . 3 ⊢ ∀:((α → ∗) → ∗)
2		alval.1	. . 3 ⊢ F:(α → ∗)
3	1, 2	wc 45	. 2 ⊢ (∀F):∗
4		df-al 116	. . 3 ⊢ ⊤⊧[∀ = λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]]
5	1, 2, 4	ceq1 79	. 2 ⊢ ⊤⊧[(∀F) = (λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]F)]
6		wv 58	. . . 4 ⊢ p:(α → ∗):(α → ∗)
7		wtru 40	. . . . 5 ⊢ ⊤:∗
8	7	wl 59	. . . 4 ⊢ λx:α ⊤:(α → ∗)
9	6, 8	weqi 68	. . 3 ⊢ [p:(α → ∗) = λx:α ⊤]:∗
10		weq 38	. . . 4 ⊢ = :((α → ∗) → ((α → ∗) → ∗))
11	6, 2	weqi 68	. . . . 5 ⊢ [p:(α → ∗) = F]:∗
12	11	id 25	. . . 4 ⊢ [p:(α → ∗) = F]⊧[p:(α → ∗) = F]
13	10, 6, 8, 12	oveq1 89	. . 3 ⊢ [p:(α → ∗) = F]⊧[[p:(α → ∗) = λx:α ⊤] = [F = λx:α ⊤]]
14	9, 2, 13	cl 106	. 2 ⊢ ⊤⊧[(λp:(α → ∗) [p:(α → ∗) = λx:α ⊤]F) = [F = λx:α ⊤]]
15	3, 5, 14	eqtri 85	1 ⊢ ⊤⊧[(∀F) = [F = λx:α ⊤]]

Syntax hints:  tv 1   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12  ∀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-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-hbl1 93  ax-17 95  ax-inst 103

This theorem depends on definitions:  df-ov 65  df-al 116

This theorem is referenced by:  ax4g  139  alrimiv  141  olc  154  orc  155  alrimi  170