Theorem beta 92

Description: Axiom of beta-substitution. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypothesis

Ref	Expression
beta.1	⊢ A:β

Assertion

Ref	Expression
beta	⊢ ⊤⊧[(λx:α Ax:α) = A]

Proof of Theorem beta

Step	Hyp	Ref	Expression
1		weq 41	. 2 ⊢ = :(β → (β → ∗))
2		beta.1	. . . 4 ⊢ A:β
3	2	wl 66	. . 3 ⊢ λx:α A:(α → β)
4		wv 64	. . 3 ⊢ x:α:α
5	3, 4	wc 50	. 2 ⊢ (λx:α Ax:α):β
6	2	ax-beta 67	. 2 ⊢ ⊤⊧(( = (λx:α Ax:α))A)
7	1, 5, 2, 6	dfov2 75	1 ⊢ ⊤⊧[(λx:α Ax:α) = A]

Syntax hints:  tv 1  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  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-wov 71

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  clf  115  ax4  150  exlimdv  167  19.8a  170  cbvf  179  leqf  181  exlimd  183  ax11  214  axrep  220