Theorem negantlem6 836

Description: Negated antecedent identity.

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 834	. . 3 (a⊥ →1 c) = (b⊥ →1 c)
3	2	negantlem5 835	. 2 (a⊥ ⊥ ∩ c⊥ ) = (b⊥ ⊥ ∩ c⊥ )
4		ax-a1 29	. . 3 a = a⊥ ⊥
5	4	ran 71	. 2 (a ∩ c⊥ ) = (a⊥ ⊥ ∩ c⊥ )
6		ax-a1 29	. . 3 b = b⊥ ⊥
7	6	ran 71	. 2 (b ∩ c⊥ ) = (b⊥ ⊥ ∩ c⊥ )
8	3, 5, 7	3tr1 60	1 (a ∩ c⊥ ) = (b ∩ c⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  negantlem8 838  negant2 840

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-le1 122  df-le2 123  df-c1 124  df-c2 125

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