Theorem negantlem8 838

Description: Negated antecedent identity.

Hypothesis

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

Assertion

Ref	Expression
negantlem8	(a⊥ ∪ c) = (b⊥ ∪ c)

Proof of Theorem negantlem8

Step	Hyp	Ref	Expression
1		negant.1	. . . 4 (a →1 c) = (b →1 c)
2	1	negantlem6 836	. . 3 (a ∩ c⊥ ) = (b ∩ c⊥ )
3	2	ax-r4 36	. 2 (a ∩ c⊥ )⊥ = (b ∩ c⊥ )⊥
4		oran2 84	. 2 (a⊥ ∪ c) = (a ∩ c⊥ )⊥
5		oran2 84	. 2 (b⊥ ∪ c) = (b ∩ c⊥ )⊥
6	3, 4, 5	3tr1 60	1 (a⊥ ∪ c) = (b⊥ ∪ c)

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

This theorem is referenced by:  negantlem9 841  negantlem10 843

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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