Theorem 2vwomr2a 364

Description: 2-variable WOML rule. (Contributed by NM, 13-Nov-1998.)

Hypothesis

Ref	Expression
2vwomr2a.1	(a →2 b) = 1

Assertion

Ref	Expression
2vwomr2a	(a →1 b) = 1

Proof of Theorem 2vwomr2a

Step	Hyp	Ref	Expression
1		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
2		df-i2 45	. . . . 5 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
3	2	ax-r1 35	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)
4		2vwomr2a.1	. . . 4 (a →2 b) = 1
5	3, 4	ax-r2 36	. . 3 (b ∪ (a⊥ ∩ b⊥ )) = 1
6	5	2vwomr2 362	. 2 (a⊥ ∪ (a ∩ b)) = 1
7	1, 6	ax-r2 36	1 (a →1 b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361

This theorem depends on definitions:  df-a 40  df-i1 44  df-i2 45

This theorem is referenced by:  lem3.4.3  1076