Theorem negant3 842

Description: Negated antecedent identity.

Hypothesis

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

Assertion

Ref	Expression
negant3	(a⊥ →3 c) = (b⊥ →3 c)

Proof of Theorem negant3

Step	Hyp	Ref	Expression
1		negant.1	. . . 4 (a →1 c) = (b →1 c)
2	1	sac 817	. . 3 (a⊥ →1 c) = (b⊥ →1 c)
3	2	negantlem9 841	. 2 (a⊥ →3 c) ≤ (b⊥ →3 c)
4	2	ax-r1 34	. . 3 (b⊥ →1 c) = (a⊥ →1 c)
5	4	negantlem9 841	. 2 (b⊥ →3 c) ≤ (a⊥ →3 c)
6	3, 5	lebi 137	1 (a⊥ →3 c) = (b⊥ →3 c)

Syntax hints:   = wb 1  ^⊥ wn 4   →₁ wi1 13   →₃ wi3 15

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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