Theorem 3vded13 798

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

Hypotheses

Ref	Expression
3vded13.1	(b ∩ (c →1 a)) ≤ (c →1 (b →1 a))
3vded13.2	c ≤ a

Assertion

Ref	Expression
3vded13	c ≤ (b →1 a)

Proof of Theorem 3vded13

Step	Hyp	Ref	Expression
1		an1 98	. . . . 5 (b ∩ 1) = b
2	1	ax-r1 34	. . . 4 b = (b ∩ 1)
3		3vded13.2	. . . . . . 7 c ≤ a
4	3	u1lemle1 692	. . . . . 6 (c →1 a) = 1
5	4	ax-r1 34	. . . . 5 1 = (c →1 a)
6	5	lan 70	. . . 4 (b ∩ 1) = (b ∩ (c →1 a))
7	2, 6	ax-r2 35	. . 3 b = (b ∩ (c →1 a))
8		3vded13.1	. . 3 (b ∩ (c →1 a)) ≤ (c →1 (b →1 a))
9	7, 8	bltr 130	. 2 b ≤ (c →1 (b →1 a))
10	9	3vded11 796	1 c ≤ (b →1 a)

Syntax hints:   ≤ wle 2   ∩ 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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