Theorem 3vded11 814

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

Hypothesis

Ref	Expression
3vded11.1	b ≤ (c →1 (b →1 a))

Assertion

Ref	Expression
3vded11	c ≤ (b →1 a)

Proof of Theorem 3vded11

Step	Hyp	Ref	Expression
1		le1 146	. . 3 (c →1 (b →1 a)) ≤ 1
2		df-t 41	. . . . 5 1 = ((b ∪ c⊥ ) ∪ (b ∪ c⊥ )⊥ )
3		ancom 74	. . . . . . . 8 (c ∩ b⊥ ) = (b⊥ ∩ c)
4		anor2 89	. . . . . . . 8 (b⊥ ∩ c) = (b ∪ c⊥ )⊥
5	3, 4	ax-r2 36	. . . . . . 7 (c ∩ b⊥ ) = (b ∪ c⊥ )⊥
6	5	lor 70	. . . . . 6 ((b ∪ c⊥ ) ∪ (c ∩ b⊥ )) = ((b ∪ c⊥ ) ∪ (b ∪ c⊥ )⊥ )
7	6	ax-r1 35	. . . . 5 ((b ∪ c⊥ ) ∪ (b ∪ c⊥ )⊥ ) = ((b ∪ c⊥ ) ∪ (c ∩ b⊥ ))
8		ax-a3 32	. . . . 5 ((b ∪ c⊥ ) ∪ (c ∩ b⊥ )) = (b ∪ (c⊥ ∪ (c ∩ b⊥ )))
9	2, 7, 8	3tr 65	. . . 4 1 = (b ∪ (c⊥ ∪ (c ∩ b⊥ )))
10		3vded11.1	. . . . 5 b ≤ (c →1 (b →1 a))
11		leo 158	. . . . . . . . 9 b⊥ ≤ (b⊥ ∪ (b ∩ a))
12		df-i1 44	. . . . . . . . . 10 (b →1 a) = (b⊥ ∪ (b ∩ a))
13	12	ax-r1 35	. . . . . . . . 9 (b⊥ ∪ (b ∩ a)) = (b →1 a)
14	11, 13	lbtr 139	. . . . . . . 8 b⊥ ≤ (b →1 a)
15	14	lelan 167	. . . . . . 7 (c ∩ b⊥ ) ≤ (c ∩ (b →1 a))
16	15	lelor 166	. . . . . 6 (c⊥ ∪ (c ∩ b⊥ )) ≤ (c⊥ ∪ (c ∩ (b →1 a)))
17		df-i1 44	. . . . . . 7 (c →1 (b →1 a)) = (c⊥ ∪ (c ∩ (b →1 a)))
18	17	ax-r1 35	. . . . . 6 (c⊥ ∪ (c ∩ (b →1 a))) = (c →1 (b →1 a))
19	16, 18	lbtr 139	. . . . 5 (c⊥ ∪ (c ∩ b⊥ )) ≤ (c →1 (b →1 a))
20	10, 19	lel2or 170	. . . 4 (b ∪ (c⊥ ∪ (c ∩ b⊥ ))) ≤ (c →1 (b →1 a))
21	9, 20	bltr 138	. . 3 1 ≤ (c →1 (b →1 a))
22	1, 21	lebi 145	. 2 (c →1 (b →1 a)) = 1
23	22	u1lemle2 715	1 c ≤ (b →1 a)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ 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:  3vded13  816