Theorem hbxfrf 107

Description: Transfer a hypothesis builder to an equivalent expression. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
hbxfr.1	⊢ T:β
hbxfr.2	⊢ B:α
hbxfrf.3	⊢ R⊧[T = A]
hbxfrf.4	⊢ (S, R)⊧[(λx:α AB) = A]

Assertion

Ref	Expression
hbxfrf	⊢ (S, R)⊧[(λx:α TB) = T]

Distinct variable group:   x,R

Proof of Theorem hbxfrf

Step	Hyp	Ref	Expression
1		hbxfr.1	. . . . 5 ⊢ T:β
2		hbxfrf.3	. . . . 5 ⊢ R⊧[T = A]
3	1, 2	eqtypi 78	. . . 4 ⊢ A:β
4	3	wl 66	. . 3 ⊢ λx:α A:(α → β)
5		hbxfr.2	. . 3 ⊢ B:α
6	4, 5	wc 50	. 2 ⊢ (λx:α AB):β
7		hbxfrf.4	. 2 ⊢ (S, R)⊧[(λx:α AB) = A]
8	1	wl 66	. . . 4 ⊢ λx:α T:(α → β)
9	1, 2	leq 91	. . . 4 ⊢ R⊧[λx:α T = λx:α A]
10	8, 5, 9	ceq1 89	. . 3 ⊢ R⊧[(λx:α TB) = (λx:α AB)]
11	7	ax-cb1 29	. . . 4 ⊢ (S, R):∗
12	11	wctl 33	. . 3 ⊢ S:∗
13	10, 12	adantl 56	. 2 ⊢ (S, R)⊧[(λx:α TB) = (λx:α AB)]
14	2, 12	adantl 56	. 2 ⊢ (S, R)⊧[T = A]
15	6, 7, 13, 14	3eqtr4i 96	1 ⊢ (S, R)⊧[(λx:α TB) = T]

Syntax hints:  kc 5  λkl 6   = ke 7  [kbr 9  kct 10  ⊧wffMMJ2 11  wffMMJ2t 12

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wl 65  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  hbxfr  108  hbov  111  hbct  155