Theorem oacom 991

Description: Commutation law requiring OA.

Hypotheses

Ref	Expression
oacom.1	d C ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
oacom.2	(d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) C (a →2 b)

Assertion

Ref	Expression
oacom	d C ((a →2 b) ∩ (a →2 c))

Proof of Theorem oacom

Step	Hyp	Ref	Expression
1		oacom.1	. . . . 5 d C ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
2	1	comcom 435	. . . 4 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) C d
3		ancom 68	. . . . . 6 (((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) ∩ d) = (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
4		oacom.2	. . . . . 6 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) C (a →2 b)
5	3, 4	bctr 173	. . . . 5 (((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) ∩ d) C (a →2 b)
6	5	comcom 435	. . . 4 (a →2 b) C (((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) ∩ d)
7	2, 6	gsth2 472	. . 3 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) C d
8	7	comcom 435	. 2 d C ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
9		df-i0 42	. . . 4 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
10	9	lan 70	. . 3 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) = ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))
11		oath1 984	. . 3 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) = ((a →2 b) ∩ (a →2 c))
12	10, 11	ax-r2 35	. 2 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) = ((a →2 b) ∩ (a →2 c))
13	8, 12	cbtr 174	1 d C ((a →2 b) ∩ (a →2 c))

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 12   →₂ wi2 14

This theorem is referenced by:  oacom2 992

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421  ax-3oa 978

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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