Theorem oacom2 992

Description: Commutation law requiring OA.

Hypothesis

Ref	Expression
oacom2.1	d ≤ ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))

Assertion

Ref	Expression
oacom2	d C ((a →2 b) ∩ (a →2 c))

Proof of Theorem oacom2

Step	Hyp	Ref	Expression
1		oacom2.1	. . . 4 d ≤ ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))))
2		lear 153	. . . 4 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
3	1, 2	letr 129	. . 3 d ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
4	3	lecom 172	. 2 d C ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
5		lea 152	. . . 4 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ d
6		lea 152	. . . . 5 ((a →2 b) ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 b)
7	1, 6	letr 129	. . . 4 d ≤ (a →2 b)
8	5, 7	letr 129	. . 3 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 b)
9	8	lecom 172	. 2 (d ∩ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))) C (a →2 b)
10	4, 9	oacom 991	1 d C ((a →2 b) ∩ (a →2 c))

Syntax hints:   ≤ wle 2   C wc 3   ∪ wo 6   ∩ wa 7   →₀ wi0 12   →₂ wi2 14

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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