Theorem negant2 840

Description: Negated antecedent identity.

Hypothesis

Ref	Expression
negant.1	(a ->1 c) = (b ->1 c)

Assertion

Ref	Expression
negant2	(a_|_ ->2 c) = (b_|_ ->2 c)

Proof of Theorem negant2

Step	Hyp	Ref	Expression
1		negant.1	. . . . 5 (a ->1 c) = (b ->1 c)
2	1	negantlem6 836	. . . 4 (a ^ c_|_) = (b ^ c_|_)
3		ax-a1 29	. . . . 5 a = a_|__|_
4	3	ran 71	. . . 4 (a ^ c_|_) = (a_|__|_ ^ c_|_)
5		ax-a1 29	. . . . 5 b = b_|__|_
6	5	ran 71	. . . 4 (b ^ c_|_) = (b_|__|_ ^ c_|_)
7	2, 4, 6	3tr2 61	. . 3 (a_|__|_ ^ c_|_) = (b_|__|_ ^ c_|_)
8	7	lor 66	. 2 (c v (a_|__|_ ^ c_|_)) = (c v (b_|__|_ ^ c_|_))
9		df-i2 44	. 2 (a_|_ ->2 c) = (c v (a_|__|_ ^ c_|_))
10		df-i2 44	. 2 (b_|_ ->2 c) = (c v (b_|__|_ ^ c_|_))
11	8, 9, 10	3tr1 60	1 (a_|_ ->2 c) = (b_|_ ->2 c)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14

This theorem is referenced by:  negant5 845

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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