Theorem beta 82

Description: Axiom of beta-substitution.

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 38	. 2 ⊢ = :(β → (β → ∗))
2		beta.1	. . . 4 ⊢ A:β
3	2	wl 59	. . 3 ⊢ λx:α A:(α → β)
4		wv 58	. . 3 ⊢ x:α:α
5	3, 4	wc 45	. 2 ⊢ (λx:α Ax:α):β
6	2	ax-beta 60	. 2 ⊢ ⊤⊧(( = (λx:α Ax:α))A)
7	1, 5, 2, 6	dfov2 67	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-refl 39  ax-eqmp 42  ax-ceq 46  ax-beta 60

This theorem depends on definitions:  df-ov 65

This theorem is referenced by:  clf  105  ax4  140  exlimdv  157  19.8a  160  cbvf  167  leqf  169  exlimd  171  ax11  201  axrep  207