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Theorem ceq12 78
Description: Equality theorem for combination.
Hypotheses
Ref Expression
ceq12.1 |- F:(al -> be)
ceq12.2 |- A:al
ceq12.3 |- R |= [F = T]
ceq12.4 |- R |= [A = B]
Assertion
Ref Expression
ceq12 |- R |= [(FA) = (TB)]
Proof of Theorem ceq12
StepHypRef Expression
1 weq 38 . 2 |- = :(be -> (be -> *))
2 ceq12.1 . . 3 |- F:(al -> be)
3 ceq12.2 . . 3 |- A:al
42, 3wc 45 . 2 |- (FA):be
5 ceq12.3 . . . 4 |- R |= [F = T]
62, 5eqtypi 69 . . 3 |- T:(al -> be)
7 ceq12.4 . . . 4 |- R |= [A = B]
83, 7eqtypi 69 . . 3 |- B:al
96, 8wc 45 . 2 |- (TB):be
10 weq 38 . . . 4 |- = :((al -> be) -> ((al -> be) -> *))
1110, 2, 6, 5dfov1 66 . . 3 |- R |= (( = F)T)
12 weq 38 . . . 4 |- = :(al -> (al -> *))
1312, 3, 8, 7dfov1 66 . . 3 |- R |= (( = A)B)
142, 6, 3, 8ax-ceq 46 . . 3 |- ((( = F)T), (( = A)B)) |= (( = (FA))(TB))
1511, 13, 14syl2anc 19 . 2 |- R |= (( = (FA))(TB))
161, 4, 9, 15dfov2 67 1 |- R |= [(FA) = (TB)]
Colors of variables: type var term
Syntax hints:   -> ht 2  kc 5   = ke 7  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ceq 46
This theorem depends on definitions:  df-ov 65
This theorem is referenced by:  ceq1  79  ceq2  80  oveq123  88  hbc  100  ac  184
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