Theorem oagen2 1016

Description: "Generalized" OA. (Contributed by NM, 19-Nov-1998.)

Hypothesis

Ref	Expression
oagen2.1	d ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))

Assertion

Ref	Expression
oagen2	((a →2 b) ∩ d) ≤ (a →2 c)

Proof of Theorem oagen2

Step	Hyp	Ref	Expression
1		oagen2.1	. . . 4 d ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
2		df-i0 43	. . . 4 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) = ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
3	1, 2	lbtr 139	. . 3 d ≤ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))
4	3	lelan 167	. 2 ((a →2 b) ∩ d) ≤ ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c))))
5		oal2 999	. 2 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)
6	4, 5	letr 137	1 ((a →2 b) ∩ d) ≤ (a →2 c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-3oa 998

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  oagen2b  1017