Theorem 3vded12 797

Description: A 3-variable theorem. Experiment with weak deduction theorem.

Hypotheses

Ref	Expression
3vded12.1	(a ∩ (c →1 a)) ≤ (c →1 (b →1 a))
3vded12.2	c ≤ a

Assertion

Ref	Expression
3vded12	c ≤ (b →1 a)

Proof of Theorem 3vded12

Step	Hyp	Ref	Expression
1		le1 138	. . 3 (c →1 (b →1 a)) ≤ 1
2		df-t 40	. . . 4 1 = (a ∪ a⊥ )
3		an1 98	. . . . . . . 8 (a ∩ 1) = a
4	3	ax-r1 34	. . . . . . 7 a = (a ∩ 1)
5		3vded12.2	. . . . . . . . . 10 c ≤ a
6	5	u1lemle1 692	. . . . . . . . 9 (c →1 a) = 1
7	6	lan 70	. . . . . . . 8 (a ∩ (c →1 a)) = (a ∩ 1)
8	7	ax-r1 34	. . . . . . 7 (a ∩ 1) = (a ∩ (c →1 a))
9	4, 8	ax-r2 35	. . . . . 6 a = (a ∩ (c →1 a))
10		3vded12.1	. . . . . 6 (a ∩ (c →1 a)) ≤ (c →1 (b →1 a))
11	9, 10	bltr 130	. . . . 5 a ≤ (c →1 (b →1 a))
12	5	lecon 146	. . . . . 6 a⊥ ≤ c⊥
13		leo 150	. . . . . . 7 c⊥ ≤ (c⊥ ∪ (c ∩ (b →1 a)))
14		df-i1 43	. . . . . . . 8 (c →1 (b →1 a)) = (c⊥ ∪ (c ∩ (b →1 a)))
15	14	ax-r1 34	. . . . . . 7 (c⊥ ∪ (c ∩ (b →1 a))) = (c →1 (b →1 a))
16	13, 15	lbtr 131	. . . . . 6 c⊥ ≤ (c →1 (b →1 a))
17	12, 16	letr 129	. . . . 5 a⊥ ≤ (c →1 (b →1 a))
18	11, 17	lel2or 162	. . . 4 (a ∪ a⊥ ) ≤ (c →1 (b →1 a))
19	2, 18	bltr 130	. . 3 1 ≤ (c →1 (b →1 a))
20	1, 19	lebi 137	. 2 (c →1 (b →1 a)) = 1
21	20	u1lemle2 697	1 c ≤ (b →1 a)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₁ wi1 13

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

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