Theorem hbct 155

Description: Hypothesis builder for context conjunction. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
hbct.1	⊢ A:∗
hbct.2	⊢ B:α
hbct.3	⊢ C:∗
hbct.4	⊢ R⊧[(λx:α AB) = A]
hbct.5	⊢ R⊧[(λx:α CB) = C]

Assertion

Ref	Expression
hbct	⊢ R⊧[(λx:α (A, C)B) = (A, C)]

Proof of Theorem hbct

Step	Hyp	Ref	Expression
1		hbct.4	. . . 4 ⊢ R⊧[(λx:α AB) = A]
2	1	ax-cb1 29	. . 3 ⊢ R:∗
3	2	trud 27	. 2 ⊢ R⊧⊤
4		hbct.1	. . . 4 ⊢ A:∗
5		hbct.3	. . . 4 ⊢ C:∗
6	4, 5	wct 48	. . 3 ⊢ (A, C):∗
7		hbct.2	. . 3 ⊢ B:α
8		wan 136	. . . . 5 ⊢ ∧ :(∗ → (∗ → ∗))
9	8, 4, 5	wov 72	. . . 4 ⊢ [A ∧ C]:∗
10	4, 5	dfan2 154	. . . 4 ⊢ ⊤⊧[[A ∧ C] = (A, C)]
11	9, 10	eqcomi 79	. . 3 ⊢ ⊤⊧[(A, C) = [A ∧ C]]
12	8, 7, 2	a17i 106	. . . . 5 ⊢ R⊧[(λx:α ∧ B) = ∧ ]
13		hbct.5	. . . . 5 ⊢ R⊧[(λx:α CB) = C]
14	8, 4, 7, 5, 12, 1, 13	hbov 111	. . . 4 ⊢ R⊧[(λx:α [A ∧ C]B) = [A ∧ C]]
15		wtru 43	. . . 4 ⊢ ⊤:∗
16	14, 15	adantr 55	. . 3 ⊢ (R, ⊤)⊧[(λx:α [A ∧ C]B) = [A ∧ C]]
17	6, 7, 11, 16	hbxfrf 107	. 2 ⊢ (R, ⊤)⊧[(λx:α (A, C)B) = (A, C)]
18	3, 17	mpdan 35	1 ⊢ R⊧[(λx:α (A, C)B) = (A, C)]

Syntax hints:   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  kct 10  ⊧wffMMJ2 11  wffMMJ2t 12   ∧ tan 119

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-ded 46  ax-wct 47  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

This theorem is referenced by:  alimdv  184  ax5  207