Theorem oaeqv 830

Description: Weakened OA implies OA). (Contributed by NM, 16-Nov-1998.)

Hypothesis

Ref	Expression
oaeqv.1	((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))

Assertion

Ref	Expression
oaeqv	((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)

Proof of Theorem oaeqv

Step	Hyp	Ref	Expression
1		lea 160	. . . 4 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 b)
2		oaeqv.1	. . . 4 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))
3	1, 2	ler2an 173	. . 3 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((a →2 b) ∩ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))
4		2oath1 826	. . 3 ((a →2 b) ∩ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))) = ((a →2 b) ∩ (a →2 c))
5	3, 4	lbtr 139	. 2 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ ((a →2 b) ∩ (a →2 c))
6		lear 161	. 2 ((a →2 b) ∩ (a →2 c)) ≤ (a →2 c)
7	5, 6	letr 137	1 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)