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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
StepHypRef Expression
1 hbxfr.3 . . . 4 R⊧[T = A]
21ax-cb1 29 . . 3 R:∗
32id 25 . 2 RR
4 hbxfr.1 . . 3 T:β
5 hbxfr.2 . . 3 B:α
6 hbxfr.4 . . . 4 R⊧[(λx:α AB) = A]
76, 2adantr 55 . . 3 (R, R)⊧[(λx:α AB) = A]
84, 5, 1, 7hbxfrf 107 . 2 (R, R)⊧[(λx:α TB) = T]
93, 3, 8syl2anc 19 1 R⊧[(λx:α TB) = T]
 Colors of variables: type var term 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
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