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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:be
hbxfr.2 |- B:al
hbxfrf.3 |- R |= [T = A]
hbxfrf.4 |- (S, R) |= [(\x:al AB) = A]
Assertion
Ref Expression
hbxfrf |- (S, R) |= [(\x:al TB) = T]
Distinct variable group:   x,R

Proof of Theorem hbxfrf
StepHypRef Expression
1 hbxfr.1 . . . . 5 |- T:be
2 hbxfrf.3 . . . . 5 |- R |= [T = A]
31, 2eqtypi 78 . . . 4 |- A:be
43wl 66 . . 3 |- \x:al A:(al -> be)
5 hbxfr.2 . . 3 |- B:al
64, 5wc 50 . 2 |- (\x:al AB):be
7 hbxfrf.4 . 2 |- (S, R) |= [(\x:al AB) = A]
81wl 66 . . . 4 |- \x:al T:(al -> be)
91, 2leq 91 . . . 4 |- R |= [\x:al T = \x:al A]
108, 5, 9ceq1 89 . . 3 |- R |= [(\x:al TB) = (\x:al AB)]
117ax-cb1 29 . . . 4 |- (S, R):*
1211wctl 33 . . 3 |- S:*
1310, 12adantl 56 . 2 |- (S, R) |= [(\x:al TB) = (\x:al AB)]
142, 12adantl 56 . 2 |- (S, R) |= [T = A]
156, 7, 13, 143eqtr4i 96 1 |- (S, R) |= [(\x:al TB) = T]
Colors of variables: type var term
Syntax hints:  kc 5  \kl 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
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