Theorem negant2 858

Description: Negated antecedent identity. (Contributed by NM, 6-Aug-2001.)

Hypothesis

Ref	Expression
negant.1	(a →1 c) = (b →1 c)

Assertion

Ref	Expression
negant2	(a⊥ →2 c) = (b⊥ →2 c)

Proof of Theorem negant2

Step	Hyp	Ref	Expression
1		negant.1	. . . . 5 (a →1 c) = (b →1 c)
2	1	negantlem6 854	. . . 4 (a ∩ c⊥ ) = (b ∩ c⊥ )
3		ax-a1 30	. . . . 5 a = a⊥ ⊥
4	3	ran 78	. . . 4 (a ∩ c⊥ ) = (a⊥ ⊥ ∩ c⊥ )
5		ax-a1 30	. . . . 5 b = b⊥ ⊥
6	5	ran 78	. . . 4 (b ∩ c⊥ ) = (b⊥ ⊥ ∩ c⊥ )
7	2, 4, 6	3tr2 64	. . 3 (a⊥ ⊥ ∩ c⊥ ) = (b⊥ ⊥ ∩ c⊥ )
8	7	lor 70	. 2 (c ∪ (a⊥ ⊥ ∩ c⊥ )) = (c ∪ (b⊥ ⊥ ∩ c⊥ ))
9		df-i2 45	. 2 (a⊥ →2 c) = (c ∪ (a⊥ ⊥ ∩ c⊥ ))
10		df-i2 45	. 2 (b⊥ →2 c) = (c ∪ (b⊥ ⊥ ∩ c⊥ ))
11	8, 9, 10	3tr1 63	1 (a⊥ →2 c) = (b⊥ →2 c)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →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-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  negant5  863