Theorem 3vded13 816

Description: A 3-variable theorem. Experiment with weak deduction theorem. (Contributed by NM, 25-Oct-1998.)

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 106	. . . . 5 (b ∩ 1) = b
2	1	ax-r1 35	. . . 4 b = (b ∩ 1)
3		3vded13.2	. . . . . . 7 c ≤ a
4	3	u1lemle1 710	. . . . . 6 (c →1 a) = 1
5	4	ax-r1 35	. . . . 5 1 = (c →1 a)
6	5	lan 77	. . . 4 (b ∩ 1) = (b ∩ (c →1 a))
7	2, 6	ax-r2 36	. . 3 b = (b ∩ (c →1 a))
8		3vded13.1	. . 3 (b ∩ (c →1 a)) ≤ (c →1 (b →1 a))
9	7, 8	bltr 138	. 2 b ≤ (c →1 (b →1 a))
10	9	3vded11 814	1 c ≤ (b →1 a)

Syntax hints:   ≤ wle 2   ∩ wa 7  1wt 8   →₁ wi1 12

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

This theorem is referenced by: (None)