Theorem negantlem5 853

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

Hypothesis

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

Assertion

Ref	Expression
negantlem5	(a⊥ ∩ c⊥ ) = (b⊥ ∩ c⊥ )

Proof of Theorem negantlem5

Step	Hyp	Ref	Expression
1		negant.1	. . 3 (a →1 c) = (b →1 c)
2	1	ran 78	. 2 ((a →1 c) ∩ c⊥ ) = ((b →1 c) ∩ c⊥ )
3		u1lemanb 615	. 2 ((a →1 c) ∩ c⊥ ) = (a⊥ ∩ c⊥ )
4		u1lemanb 615	. 2 ((b →1 c) ∩ c⊥ ) = (b⊥ ∩ c⊥ )
5	2, 3, 4	3tr2 64	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:  negantlem6  854  negantlem7  855  negantlem9  859