Theorem hbxfr 108

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
hbxfr.3	⊢ R⊧[T = A]
hbxfr.4	⊢ R⊧[(λx:α AB) = A]

Assertion

Ref	Expression
hbxfr	⊢ R⊧[(λx:α TB) = T]

Distinct variable group:   x,R

Proof of Theorem hbxfr

Step	Hyp	Ref	Expression
1		hbxfr.3	. . . 4 ⊢ R⊧[T = A]
2	1	ax-cb1 29	. . 3 ⊢ R:∗
3	2	id 25	. 2 ⊢ R⊧R
4		hbxfr.1	. . 3 ⊢ T:β
5		hbxfr.2	. . 3 ⊢ B:α
6		hbxfr.4	. . . 4 ⊢ R⊧[(λx:α AB) = A]
7	6, 2	adantr 55	. . 3 ⊢ (R, R)⊧[(λx:α AB) = A]
8	4, 5, 1, 7	hbxfrf 107	. 2 ⊢ (R, R)⊧[(λx:α TB) = T]
9	3, 3, 8	syl2anc 19	1 ⊢ R⊧[(λx:α TB) = T]

Syntax hints:  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-simpl 20  ax-simpr 21  ax-id 24  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:  hbth  109