Theorem hbxfrf 97

Description: Transfer a hypothesis builder to an equivalent expression.

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 69	. . . 4 ⊢ A:β
4	3	wl 59	. . 3 ⊢ λx:α A:(α → β)
5		hbxfr.2	. . 3 ⊢ B:α
6	4, 5	wc 45	. 2 ⊢ (λx:α AB):β
7		hbxfrf.4	. 2 ⊢ (S, R)⊧[(λx:α AB) = A]
8	1	wl 59	. . . 4 ⊢ λx:α T:(α → β)
9	1, 2	leq 81	. . . 4 ⊢ R⊧[λx:α T = λx:α A]
10	8, 5, 9	ceq1 79	. . 3 ⊢ R⊧[(λx:α TB) = (λx:α AB)]
11	7	ax-cb1 29	. . . 4 ⊢ (S, R):∗
12	11	wctl 31	. . 3 ⊢ S:∗
13	10, 12	adantl 51	. 2 ⊢ (S, R)⊧[(λx:α TB) = (λx:α AB)]
14	2, 12	adantl 51	. 2 ⊢ (S, R)⊧[T = A]
15	6, 7, 13, 14	3eqtr4i 86	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-refl 39  ax-eqmp 42  ax-ceq 46  ax-leq 62

This theorem depends on definitions:  df-ov 65

This theorem is referenced by:  hbxfr  98  hbov  101  hbct  145