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	⊢ ⊤⊧[[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 ⊢ ⊤⊧[ ⇒ = λp:∗ λq:∗ [[p:∗ ∧ q:∗] = p:∗]]
6	1, 2, 3, 5	oveq 102	. 2 ⊢ ⊤⊧[[A ⇒ B] = [Aλp:∗ λq:∗ [[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 ⊢ ⊤⊧[[Aλp:∗ λq:∗ [[p:∗ ∧ q:∗] = p:∗]B] = [[A ∧ B] = A]]
23	4, 6, 22	eqtri 95	1 ⊢ ⊤⊧[[A ⇒ B] = [[A ∧ B] = A]]

This proof depends on syntax axioms:  tv 1  ∗hb 3  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12   ∧ tan 119   ⇒ tim 121

This proof depends on 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 proof depends on definitions:  df-ov 73  df-an 128  df-im 129

This theorem is used by:  mpd  156  ex  158