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Theorem hbct 145
Description: Hypothesis builder for context conjunction.
Hypotheses
Ref Expression
hbct.1 |- A:*
hbct.2 |- B:al
hbct.3 |- C:*
hbct.4 |- R |= [(\x:al AB) = A]
hbct.5 |- R |= [(\x:al CB) = C]
Assertion
Ref Expression
hbct |- R |= [(\x:al (A, C)B) = (A, C)]
Proof of Theorem hbct
StepHypRef Expression
1 hbct.4 . . . 4 |- R |= [(\x:al AB) = A]
21ax-cb1 29 . . 3 |- R:*
32trud 27 . 2 |- R |= T.
4 hbct.1 . . . 4 |- A:*
5 hbct.3 . . . 4 |- C:*
64, 5wct 44 . . 3 |- (A, C):*
7 hbct.2 . . 3 |- B:al
8 wan 126 . . . . 5 |- /\ :(* -> (* -> *))
98, 4, 5wov 64 . . . 4 |- [A /\ C]:*
104, 5dfan2 144 . . . 4 |- T. |= [[A /\ C] = (A, C)]
119, 10eqcomi 70 . . 3 |- T. |= [(A, C) = [A /\ C]]
128, 7, 2a17i 96 . . . . 5 |- R |= [(\x:al /\ B) = /\ ]
13 hbct.5 . . . . 5 |- R |= [(\x:al CB) = C]
148, 4, 7, 5, 12, 1, 13hbov 101 . . . 4 |- R |= [(\x:al [A /\ C]B) = [A /\ C]]
15 wtru 40 . . . 4 |- T.:*
1614, 15adantr 50 . . 3 |- (R, T.) |= [(\x:al [A /\ C]B) = [A /\ C]]
176, 7, 11, 16hbxfrf 97 . 2 |- (R, T.) |= [(\x:al (A, C)B) = (A, C)]
183, 17mpdan 33 1 |- R |= [(\x:al (A, C)B) = (A, C)]
Colors of variables: type var term
Syntax hints:   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   /\ tan 109
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-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103
This theorem depends on definitions:  df-ov 65  df-an 118
This theorem is referenced by:  alimdv  172  ax5  194
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