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Theorem anval 138
Description: Value of the conjunction.
Hypotheses
Ref Expression
imval.1 |- A:*
imval.2 |- B:*
Assertion
Ref Expression
anval |- T. |= [[A /\ B] = [\f:(* -> (* -> *)) [Af:(* -> (* -> *))B] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
Distinct variable groups:   f,A   f,B
Proof of Theorem anval
Dummy variables p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wan 126 . . 3 |- /\ :(* -> (* -> *))
2 imval.1 . . 3 |- A:*
3 imval.2 . . 3 |- B:*
41, 2, 3wov 64 . 2 |- [A /\ B]:*
5 df-an 118 . . 3 |- T. |= [ /\ = \p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
61, 2, 3, 5oveq 92 . 2 |- T. |= [[A /\ B] = [A\p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]B]]
7 wv 58 . . . . . 6 |- f:(* -> (* -> *)):(* -> (* -> *))
8 wv 58 . . . . . 6 |- p:*:*
9 wv 58 . . . . . 6 |- q:*:*
107, 8, 9wov 64 . . . . 5 |- [p:*f:(* -> (* -> *))q:*]:*
1110wl 59 . . . 4 |- \f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*]:((* -> (* -> *)) -> *)
12 wtru 40 . . . . . 6 |- T.:*
137, 12, 12wov 64 . . . . 5 |- [T.f:(* -> (* -> *))T.]:*
1413wl 59 . . . 4 |- \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]:((* -> (* -> *)) -> *)
1511, 14weqi 68 . . 3 |- [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]:*
16 weq 38 . . . 4 |- = :(((* -> (* -> *)) -> *) -> (((* -> (* -> *)) -> *) -> *))
178, 2weqi 68 . . . . . . 7 |- [p:* = A]:*
1817id 25 . . . . . 6 |- [p:* = A] |= [p:* = A]
197, 8, 9, 18oveq1 89 . . . . 5 |- [p:* = A] |= [[p:*f:(* -> (* -> *))q:*] = [Af:(* -> (* -> *))q:*]]
2010, 19leq 81 . . . 4 |- [p:* = A] |= [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [Af:(* -> (* -> *))q:*]]
2116, 11, 14, 20oveq1 89 . . 3 |- [p:* = A] |= [[\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]] = [\f:(* -> (* -> *)) [Af:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
227, 2, 9wov 64 . . . . 5 |- [Af:(* -> (* -> *))q:*]:*
2322wl 59 . . . 4 |- \f:(* -> (* -> *)) [Af:(* -> (* -> *))q:*]:((* -> (* -> *)) -> *)
249, 3weqi 68 . . . . . . 7 |- [q:* = B]:*
2524id 25 . . . . . 6 |- [q:* = B] |= [q:* = B]
267, 2, 9, 25oveq2 91 . . . . 5 |- [q:* = B] |= [[Af:(* -> (* -> *))q:*] = [Af:(* -> (* -> *))B]]
2722, 26leq 81 . . . 4 |- [q:* = B] |= [\f:(* -> (* -> *)) [Af:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [Af:(* -> (* -> *))B]]
2816, 23, 14, 27oveq1 89 . . 3 |- [q:* = B] |= [[\f:(* -> (* -> *)) [Af:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]] = [\f:(* -> (* -> *)) [Af:(* -> (* -> *))B] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
2915, 2, 3, 21, 28ovl 107 . 2 |- T. |= [[A\p:* \q:* [\f:(* -> (* -> *)) [p:*f:(* -> (* -> *))q:*] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]B] = [\f:(* -> (* -> *)) [Af:(* -> (* -> *))B] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
304, 6, 29eqtri 85 1 |- T. |= [[A /\ B] = [\f:(* -> (* -> *)) [Af:(* -> (* -> *))B] = \f:(* -> (* -> *)) [T.f:(* -> (* -> *))T.]]]
Colors of variables: type var term
Syntax hints:  tv 1   -> ht 2  *hb 3  \kl 6   = ke 7  T.kt 8  [kbr 9   |= 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-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:  dfan2  144
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