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Theorem oveq123 88
Description: Equality theorem for binary operation.
Hypotheses
Ref Expression
oveq.1 |- F:(al -> (be -> ga))
oveq.2 |- A:al
oveq.3 |- B:be
oveq123.4 |- R |= [F = S]
oveq123.5 |- R |= [A = C]
oveq123.6 |- R |= [B = T]
Assertion
Ref Expression
oveq123 |- R |= [[AFB] = [CST]]

Proof of Theorem oveq123
StepHypRef Expression
1 oveq.1 . . . 4 |- F:(al -> (be -> ga))
2 oveq.2 . . . 4 |- A:al
31, 2wc 45 . . 3 |- (FA):(be -> ga)
4 oveq.3 . . 3 |- B:be
53, 4wc 45 . 2 |- ((FA)B):ga
6 oveq123.4 . . . 4 |- R |= [F = S]
7 oveq123.5 . . . 4 |- R |= [A = C]
81, 2, 6, 7ceq12 78 . . 3 |- R |= [(FA) = (SC)]
9 oveq123.6 . . 3 |- R |= [B = T]
103, 4, 8, 9ceq12 78 . 2 |- R |= [((FA)B) = ((SC)T)]
11 weq 38 . . 3 |- = :(ga -> (ga -> *))
121, 2, 4wov 64 . . 3 |- [AFB]:ga
136ax-cb1 29 . . . 4 |- R:*
141, 2, 4df-ov 65 . . . 4 |- T. |= (( = [AFB])((FA)B))
1513, 14a1i 28 . . 3 |- R |= (( = [AFB])((FA)B))
1611, 12, 5, 15dfov2 67 . 2 |- R |= [[AFB] = ((FA)B)]
171, 6eqtypi 69 . . . 4 |- S:(al -> (be -> ga))
182, 7eqtypi 69 . . . 4 |- C:al
194, 9eqtypi 69 . . . 4 |- T:be
2017, 18, 19wov 64 . . 3 |- [CST]:ga
2117, 18wc 45 . . . 4 |- (SC):(be -> ga)
2221, 19wc 45 . . 3 |- ((SC)T):ga
2317, 18, 19df-ov 65 . . . 4 |- T. |= (( = [CST])((SC)T))
2413, 23a1i 28 . . 3 |- R |= (( = [CST])((SC)T))
2511, 20, 22, 24dfov2 67 . 2 |- R |= [[CST] = ((SC)T)]
265, 10, 16, 253eqtr4i 86 1 |- R |= [[AFB] = [CST]]
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:  oveq1  89  oveq12  90  oveq  92
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