Theorem wan 136

Description: Conjunction type. (Contributed by Mario Carneiro, 8-Oct-2014.)

Assertion

Ref	Expression
wan	⊢ ∧ :(∗ → (∗ → ∗))

Proof of Theorem wan

Dummy variables f p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wv 64	. . . . . . 7 ⊢ f:(∗ → (∗ → ∗)):(∗ → (∗ → ∗))
2		wv 64	. . . . . . 7 ⊢ p:∗:∗
3		wv 64	. . . . . . 7 ⊢ q:∗:∗
4	1, 2, 3	wov 72	. . . . . 6 ⊢ [p:∗f:(∗ → (∗ → ∗))q:∗]:∗
5	4	wl 66	. . . . 5 ⊢ λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗]:((∗ → (∗ → ∗)) → ∗)
6		wtru 43	. . . . . . 7 ⊢ ⊤:∗
7	1, 6, 6	wov 72	. . . . . 6 ⊢ [⊤f:(∗ → (∗ → ∗))⊤]:∗
8	7	wl 66	. . . . 5 ⊢ λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]:((∗ → (∗ → ∗)) → ∗)
9	5, 8	weqi 76	. . . 4 ⊢ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:∗
10	9	wl 66	. . 3 ⊢ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:(∗ → ∗)
11	10	wl 66	. 2 ⊢ λp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:(∗ → (∗ → ∗))
12		df-an 128	. 2 ⊢ ⊤⊧[ ∧ = λp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
13	11, 12	eqtypri 81	1 ⊢ ∧ :(∗ → (∗ → ∗))

Syntax hints:  tv 1   → ht 2  ∗hb 3  λkl 6   = ke 7  ⊤kt 8  [kbr 9  wffMMJ2t 12   ∧ tan 119

This theorem was proved from axioms:  ax-cb1 29  ax-weq 40  ax-refl 42  ax-wv 63  ax-wl 65  ax-wov 71  ax-eqtypri 80

This theorem depends on definitions:  df-an 128

This theorem is referenced by:  wim  137  imval  146  anval  148  dfan2  154  hbct  155  ex  158  axrep  220  axun  222