Theorem hbl 112

Description: Hypothesis builder for lambda abstraction. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
hbl.1	⊢ A:γ
hbl.2	⊢ B:α
hbl.3	⊢ R⊧[(λx:α AB) = A]

Assertion

Ref	Expression
hbl	⊢ R⊧[(λx:α λy:β AB) = λy:β A]

Distinct variable groups:   x,y   y,B   y,R

Proof of Theorem hbl

Step	Hyp	Ref	Expression
1		hbl.1	. . . . 5 ⊢ A:γ
2	1	wl 66	. . . 4 ⊢ λy:β A:(β → γ)
3	2	wl 66	. . 3 ⊢ λx:α λy:β A:(α → (β → γ))
4		hbl.2	. . 3 ⊢ B:α
5	3, 4	wc 50	. 2 ⊢ (λx:α λy:β AB):(β → γ)
6		hbl.3	. . . 4 ⊢ R⊧[(λx:α AB) = A]
7	6	ax-cb1 29	. . 3 ⊢ R:∗
8	1, 4	distrl 94	. . 3 ⊢ ⊤⊧[(λx:α λy:β AB) = λy:β (λx:α AB)]
9	7, 8	a1i 28	. 2 ⊢ R⊧[(λx:α λy:β AB) = λy:β (λx:α AB)]
10	1	wl 66	. . . 4 ⊢ λx:α A:(α → γ)
11	10, 4	wc 50	. . 3 ⊢ (λx:α AB):γ
12	11, 6	leq 91	. 2 ⊢ R⊧[λy:β (λx:α AB) = λy:β A]
13	5, 9, 12	eqtri 95	1 ⊢ R⊧[(λx:α λy:β AB) = λy:β A]

Syntax hints:   → ht 2  kc 5  λkl 6   = ke 7  [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-wl 65  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  cbvf  179  ax7  209  axrep  220