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.

Step	Hyp	Ref	Expression
1		wim 137	. . 3 |- ==> :(* -> (* -> *))
2		imval.1	. . 3 |- A:*
3		imval.2	. . 3 |- B:*
4	1, 2, 3	wov 72	. 2 |- [A ==> B]:*
5		df-im 129	. . 3 |- T. |= [ ==> = \p:* \q:* [[p:* /\ q:*] = p:*]]
6	1, 2, 3, 5	oveq 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:*:*
10	7, 8, 9	wov 72	. . . 4 |- [p:* /\ q:*]:*
11	10, 8	weqi 76	. . 3 |- [[p:* /\ q:*] = p:*]:*
12		weq 41	. . . 4 |- = :(* -> (* -> *))
13	8, 2	weqi 76	. . . . . 6 |- [p:* = A]:*
14	13	id 25	. . . . 5 |- [p:* = A] |= [p:* = A]
15	7, 8, 9, 14	oveq1 99	. . . 4 |- [p:* = A] |= [[p:* /\ q:*] = [A /\ q:*]]
16	12, 10, 8, 15, 14	oveq12 100	. . 3 |- [p:* = A] |= [[[p:* /\ q:*] = p:*] = [[A /\ q:*] = A]]
17	7, 2, 9	wov 72	. . . 4 |- [A /\ q:*]:*
18	9, 3	weqi 76	. . . . . 6 |- [q:* = B]:*
19	18	id 25	. . . . 5 |- [q:* = B] |= [q:* = B]
20	7, 2, 9, 19	oveq2 101	. . . 4 |- [q:* = B] |= [[A /\ q:*] = [A /\ B]]
21	12, 17, 2, 20	oveq1 99	. . 3 |- [q:* = B] |= [[[A /\ q:*] = A] = [[A /\ B] = A]]
22	11, 2, 3, 16, 21	ovl 117	. 2 |- T. |= [[A\p:* \q:* [[p:* /\ q:*] = p:*]B] = [[A /\ B] = A]]
23	4, 6, 22	eqtri 95	1 |- T. |= [[A ==> B] = [[A /\ B] = A]]

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