Theorem negant2 840

Description: Negated antecedent identity.

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 836	. . . 4 (a ∩ c⊥ ) = (b ∩ c⊥ )
3		ax-a1 29	. . . . 5 a = a⊥ ⊥
4	3	ran 71	. . . 4 (a ∩ c⊥ ) = (a⊥ ⊥ ∩ c⊥ )
5		ax-a1 29	. . . . 5 b = b⊥ ⊥
6	5	ran 71	. . . 4 (b ∩ c⊥ ) = (b⊥ ⊥ ∩ c⊥ )
7	2, 4, 6	3tr2 61	. . 3 (a⊥ ⊥ ∩ c⊥ ) = (b⊥ ⊥ ∩ c⊥ )
8	7	lor 66	. 2 (c ∪ (a⊥ ⊥ ∩ c⊥ )) = (c ∪ (b⊥ ⊥ ∩ c⊥ ))
9		df-i2 44	. 2 (a⊥ →2 c) = (c ∪ (a⊥ ⊥ ∩ c⊥ ))
10		df-i2 44	. 2 (b⊥ →2 c) = (c ∪ (b⊥ ⊥ ∩ c⊥ ))
11	8, 9, 10	3tr1 60	1 (a⊥ →2 c) = (b⊥ →2 c)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14

This theorem is referenced by:  negant5 845

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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