Theorem negant4 862

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

Hypothesis

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

Assertion

Ref	Expression
negant4	(a⊥ →4 c) = (b⊥ →4 c)

Proof of Theorem negant4

Step	Hyp	Ref	Expression
1		negant.1	. . . 4 (a →1 c) = (b →1 c)
2	1	sac 835	. . 3 (a⊥ →1 c) = (b⊥ →1 c)
3	2	negantlem10 861	. 2 (a⊥ →4 c) ≤ (b⊥ →4 c)
4	2	ax-r1 35	. . 3 (b⊥ →1 c) = (a⊥ →1 c)
5	4	negantlem10 861	. 2 (b⊥ →4 c) ≤ (a⊥ →4 c)
6	3, 5	lebi 145	1 (a⊥ →4 c) = (b⊥ →4 c)

Syntax hints:   = wb 1  ^⊥ wn 4   →₁ wi1 12   →₄ wi4 15

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-i3 46  df-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  negant5  863