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Theorem imval 146
Description: Value of the implication. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypotheses
Ref Expression
imval.1 |- A:*
imval.2 |- B:*
Assertion
Ref Expression
imval |- T. |= [[A ==> B] = [[A /\ B] = A]]

Proof of Theorem imval
Dummy variables p q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wim 137 . . 3 |- ==> :(* -> (* -> *))
2 imval.1 . . 3 |- A:*
3 imval.2 . . 3 |- B:*
41, 2, 3wov 72 . 2 |- [A ==> B]:*
5 df-im 129 . . 3 |- T. |= [ ==> = \p:* \q:* [[p:* /\ q:*] = p:*]]
61, 2, 3, 5oveq 102 . 2 |- T. |= [[A ==> B] = [A\p:* \q:* [[p:* /\ q:*] = p:*]B]]
7 wan 136 . . . . 5 |- /\ :(* -> (* -> *))
8 wv 64 . . . . 5 |- p:*:*
9 wv 64 . . . . 5 |- q:*:*
107, 8, 9wov 72 . . . 4 |- [p:* /\ q:*]:*
1110, 8weqi 76 . . 3 |- [[p:* /\ q:*] = p:*]:*
12 weq 41 . . . 4 |- = :(* -> (* -> *))
138, 2weqi 76 . . . . . 6 |- [p:* = A]:*
1413id 25 . . . . 5 |- [p:* = A] |= [p:* = A]
157, 8, 9, 14oveq1 99 . . . 4 |- [p:* = A] |= [[p:* /\ q:*] = [A /\ q:*]]
1612, 10, 8, 15, 14oveq12 100 . . 3 |- [p:* = A] |= [[[p:* /\ q:*] = p:*] = [[A /\ q:*] = A]]
177, 2, 9wov 72 . . . 4 |- [A /\ q:*]:*
189, 3weqi 76 . . . . . 6 |- [q:* = B]:*
1918id 25 . . . . 5 |- [q:* = B] |= [q:* = B]
207, 2, 9, 19oveq2 101 . . . 4 |- [q:* = B] |= [[A /\ q:*] = [A /\ B]]
2112, 17, 2, 20oveq1 99 . . 3 |- [q:* = B] |= [[[A /\ q:*] = A] = [[A /\ B] = A]]
2211, 2, 3, 16, 21ovl 117 . 2 |- T. |= [[A\p:* \q:* [[p:* /\ q:*] = p:*]B] = [[A /\ B] = A]]
234, 6, 22eqtri 95 1 |- T. |= [[A ==> B] = [[A /\ B] = A]]
Colors of variables: type var term
Syntax hints:  tv 1  *hb 3  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   /\ tan 119   ==> tim 121
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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113
This theorem depends on definitions:  df-ov 73  df-an 128  df-im 129
This theorem is referenced by:  mpd  156  ex  158
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