Theorem negantlem6 854

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

Hypothesis

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

Assertion

Ref	Expression
negantlem6	(a ∩ c⊥ ) = (b ∩ c⊥ )

Proof of Theorem negantlem6

Step	Hyp	Ref	Expression
1		negant.1	. . . 4 (a →1 c) = (b →1 c)
2	1	negant 852	. . 3 (a⊥ →1 c) = (b⊥ →1 c)
3	2	negantlem5 853	. 2 (a⊥ ⊥ ∩ c⊥ ) = (b⊥ ⊥ ∩ c⊥ )
4		ax-a1 30	. . 3 a = a⊥ ⊥
5	4	ran 78	. 2 (a ∩ c⊥ ) = (a⊥ ⊥ ∩ c⊥ )
6		ax-a1 30	. . 3 b = b⊥ ⊥
7	6	ran 78	. 2 (b ∩ c⊥ ) = (b⊥ ⊥ ∩ c⊥ )
8	3, 5, 7	3tr1 63	1 (a ∩ c⊥ ) = (b ∩ c⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∩ wa 7   →₁ wi1 12

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-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  negantlem8  856  negant2  858