Theorem neg3antlem1 846

Description: Lemma for negated antecedent identity.

Hypothesis

Ref	Expression
neg3ant.1	(a ->3 c) = (b ->3 c)

Assertion

Ref	Expression
neg3antlem1	(a ^ c) =< (b ->1 c)

Proof of Theorem neg3antlem1

Step	Hyp	Ref	Expression
1		leo 150	. . 3 (a ^ c) =< ((a ^ c) v (a_|_ ^ c))
2		neg3ant.1	. . . . . 6 (a ->3 c) = (b ->3 c)
3	2	ran 71	. . . . 5 ((a ->3 c) ^ c) = ((b ->3 c) ^ c)
4		u3lemab 594	. . . . 5 ((a ->3 c) ^ c) = ((a ^ c) v (a_|_ ^ c))
5		u3lemab 594	. . . . 5 ((b ->3 c) ^ c) = ((b ^ c) v (b_|_ ^ c))
6	3, 4, 5	3tr2 61	. . . 4 ((a ^ c) v (a_|_ ^ c)) = ((b ^ c) v (b_|_ ^ c))
7		u1lemab 592	. . . . 5 ((b ->1 c) ^ c) = ((b ^ c) v (b_|_ ^ c))
8	7	ax-r1 34	. . . 4 ((b ^ c) v (b_|_ ^ c)) = ((b ->1 c) ^ c)
9	6, 8	ax-r2 35	. . 3 ((a ^ c) v (a_|_ ^ c)) = ((b ->1 c) ^ c)
10	1, 9	lbtr 131	. 2 (a ^ c) =< ((b ->1 c) ^ c)
11		lea 152	. 2 ((b ->1 c) ^ c) =< (b ->1 c)
12	10, 11	letr 129	1 (a ^ c) =< (b ->1 c)

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->3 wi3 15

This theorem is referenced by:  neg3ant1 848

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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