Theorem neg3ant1 848

Description: Lemma for negated antecedent identity.

Hypothesis

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

Assertion

Ref	Expression
neg3ant1	(a →1 c) = (b →1 c)

Proof of Theorem neg3ant1

Step	Hyp	Ref	Expression
1		neg3ant.1	. . . . . 6 (a →3 c) = (b →3 c)
2	1	neg3antlem2 847	. . . . 5 a⊥ ≤ (b →1 c)
3	1	neg3antlem1 846	. . . . 5 (a ∩ c) ≤ (b →1 c)
4	2, 3	lel2or 162	. . . 4 (a⊥ ∪ (a ∩ c)) ≤ (b →1 c)
5		df-i1 43	. . . 4 (b →1 c) = (b⊥ ∪ (b ∩ c))
6	4, 5	lbtr 131	. . 3 (a⊥ ∪ (a ∩ c)) ≤ (b⊥ ∪ (b ∩ c))
7	1	ax-r1 34	. . . . . 6 (b →3 c) = (a →3 c)
8	7	neg3antlem2 847	. . . . 5 b⊥ ≤ (a →1 c)
9	7	neg3antlem1 846	. . . . 5 (b ∩ c) ≤ (a →1 c)
10	8, 9	lel2or 162	. . . 4 (b⊥ ∪ (b ∩ c)) ≤ (a →1 c)
11		df-i1 43	. . . 4 (a →1 c) = (a⊥ ∪ (a ∩ c))
12	10, 11	lbtr 131	. . 3 (b⊥ ∪ (b ∩ c)) ≤ (a⊥ ∪ (a ∩ c))
13	6, 12	lebi 137	. 2 (a⊥ ∪ (a ∩ c)) = (b⊥ ∪ (b ∩ c))
14	13, 11, 5	3tr1 60	1 (a →1 c) = (b →1 c)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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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