Theorem negantlem7 855

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

Hypothesis

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

Assertion

Ref	Expression
negantlem7	(a ∪ c) = (b ∪ c)

Proof of Theorem negantlem7

Step	Hyp	Ref	Expression
1		negant.1	. . . 4 (a →1 c) = (b →1 c)
2	1	negantlem5 853	. . 3 (a⊥ ∩ c⊥ ) = (b⊥ ∩ c⊥ )
3		anor3 90	. . 3 (a⊥ ∩ c⊥ ) = (a ∪ c)⊥
4		anor3 90	. . 3 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
5	2, 3, 4	3tr2 64	. 2 (a ∪ c)⊥ = (b ∪ c)⊥
6	5	con1 66	1 (a ∪ c) = (b ∪ c)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ 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:  negant0  857