Theorem neg3antlem1 864

Description: Lemma for negated antecedent identity. (Contributed by NM, 7-Aug-2001.)

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 158	. . 3 (a ∩ c) ≤ ((a ∩ c) ∪ (a⊥ ∩ c))
2		neg3ant.1	. . . . . 6 (a →3 c) = (b →3 c)
3	2	ran 78	. . . . 5 ((a →3 c) ∩ c) = ((b →3 c) ∩ c)
4		u3lemab 612	. . . . 5 ((a →3 c) ∩ c) = ((a ∩ c) ∪ (a⊥ ∩ c))
5		u3lemab 612	. . . . 5 ((b →3 c) ∩ c) = ((b ∩ c) ∪ (b⊥ ∩ c))
6	3, 4, 5	3tr2 64	. . . 4 ((a ∩ c) ∪ (a⊥ ∩ c)) = ((b ∩ c) ∪ (b⊥ ∩ c))
7		u1lemab 610	. . . . 5 ((b →1 c) ∩ c) = ((b ∩ c) ∪ (b⊥ ∩ c))
8	7	ax-r1 35	. . . 4 ((b ∩ c) ∪ (b⊥ ∩ c)) = ((b →1 c) ∩ c)
9	6, 8	ax-r2 36	. . 3 ((a ∩ c) ∪ (a⊥ ∩ c)) = ((b →1 c) ∩ c)
10	1, 9	lbtr 139	. 2 (a ∩ c) ≤ ((b →1 c) ∩ c)
11		lea 160	. 2 ((b →1 c) ∩ c) ≤ (b →1 c)
12	10, 11	letr 137	1 (a ∩ c) ≤ (b →1 c)

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₃ wi3 14

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-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  neg3ant1  866