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:be
hbxfr.2	|- B:al
hbxfr.3	|- R |= [T = A]
hbxfr.4	|- R |= [(\x:al AB) = A]

Assertion

Ref	Expression
hbxfr	|- R |= [(\x:al 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:be
5		hbxfr.2	. . 3 |- B:al
6		hbxfr.4	. . . 4 |- R |= [(\x:al AB) = A]
7	6, 2	adantr 55	. . 3 |- (R, R) |= [(\x:al AB) = A]
8	4, 5, 1, 7	hbxfrf 107	. 2 |- (R, R) |= [(\x:al TB) = T]
9	3, 3, 8	syl2anc 19	1 |- R |= [(\x:al TB) = T]

Syntax hints:  kc 5  \kl 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