Theorem elimcons2 869

Description: Consequent elimination law. (Contributed by NM, 12-Mar-2002.)

Hypotheses

Ref	Expression
elimcons2.1	(a →1 c) = (b →1 c)
elimcons2.2	(a ∩ (c ∩ (b →1 c))) ≤ (b ∪ (c⊥ ∪ (a →1 c)⊥ ))

Assertion

Ref	Expression
elimcons2	a ≤ b

Proof of Theorem elimcons2

Step	Hyp	Ref	Expression
1		elimcons2.1	. 2 (a →1 c) = (b →1 c)
2		elimcons2.2	. . 3 (a ∩ (c ∩ (b →1 c))) ≤ (b ∪ (c⊥ ∪ (a →1 c)⊥ ))
3	1	ax-r1 35	. . . . . . 7 (b →1 c) = (a →1 c)
4		df-i1 44	. . . . . . 7 (a →1 c) = (a⊥ ∪ (a ∩ c))
5	3, 4	ax-r2 36	. . . . . 6 (b →1 c) = (a⊥ ∪ (a ∩ c))
6	5	lan 77	. . . . 5 (c ∩ (b →1 c)) = (c ∩ (a⊥ ∪ (a ∩ c)))
7	6	lan 77	. . . 4 (a ∩ (c ∩ (b →1 c))) = (a ∩ (c ∩ (a⊥ ∪ (a ∩ c))))
8		anass 76	. . . . 5 ((a ∩ c) ∩ (a⊥ ∪ (a ∩ c))) = (a ∩ (c ∩ (a⊥ ∪ (a ∩ c))))
9	8	ax-r1 35	. . . 4 (a ∩ (c ∩ (a⊥ ∪ (a ∩ c)))) = ((a ∩ c) ∩ (a⊥ ∪ (a ∩ c)))
10		leor 159	. . . . 5 (a ∩ c) ≤ (a⊥ ∪ (a ∩ c))
11	10	df2le2 136	. . . 4 ((a ∩ c) ∩ (a⊥ ∪ (a ∩ c))) = (a ∩ c)
12	7, 9, 11	3tr 65	. . 3 (a ∩ (c ∩ (b →1 c))) = (a ∩ c)
13	1	ax-r4 37	. . . . . . . 8 (a →1 c)⊥ = (b →1 c)⊥
14		ud1lem0c 277	. . . . . . . 8 (b →1 c)⊥ = (b ∩ (b⊥ ∪ c⊥ ))
15	13, 14	ax-r2 36	. . . . . . 7 (a →1 c)⊥ = (b ∩ (b⊥ ∪ c⊥ ))
16	15	lor 70	. . . . . 6 (c⊥ ∪ (a →1 c)⊥ ) = (c⊥ ∪ (b ∩ (b⊥ ∪ c⊥ )))
17		ax-a2 31	. . . . . 6 (c⊥ ∪ (b ∩ (b⊥ ∪ c⊥ ))) = ((b ∩ (b⊥ ∪ c⊥ )) ∪ c⊥ )
18	16, 17	ax-r2 36	. . . . 5 (c⊥ ∪ (a →1 c)⊥ ) = ((b ∩ (b⊥ ∪ c⊥ )) ∪ c⊥ )
19	18	lor 70	. . . 4 (b ∪ (c⊥ ∪ (a →1 c)⊥ )) = (b ∪ ((b ∩ (b⊥ ∪ c⊥ )) ∪ c⊥ ))
20		ax-a3 32	. . . . 5 ((b ∪ (b ∩ (b⊥ ∪ c⊥ ))) ∪ c⊥ ) = (b ∪ ((b ∩ (b⊥ ∪ c⊥ )) ∪ c⊥ ))
21	20	ax-r1 35	. . . 4 (b ∪ ((b ∩ (b⊥ ∪ c⊥ )) ∪ c⊥ )) = ((b ∪ (b ∩ (b⊥ ∪ c⊥ ))) ∪ c⊥ )
22		ax-a2 31	. . . . . 6 (b ∪ (b ∩ (b⊥ ∪ c⊥ ))) = ((b ∩ (b⊥ ∪ c⊥ )) ∪ b)
23		lea 160	. . . . . . 7 (b ∩ (b⊥ ∪ c⊥ )) ≤ b
24	23	df-le2 131	. . . . . 6 ((b ∩ (b⊥ ∪ c⊥ )) ∪ b) = b
25	22, 24	ax-r2 36	. . . . 5 (b ∪ (b ∩ (b⊥ ∪ c⊥ ))) = b
26	25	ax-r5 38	. . . 4 ((b ∪ (b ∩ (b⊥ ∪ c⊥ ))) ∪ c⊥ ) = (b ∪ c⊥ )
27	19, 21, 26	3tr 65	. . 3 (b ∪ (c⊥ ∪ (a →1 c)⊥ )) = (b ∪ c⊥ )
28	2, 12, 27	le3tr2 141	. 2 (a ∩ c) ≤ (b ∪ c⊥ )
29	1, 28	elimcons 868	1 a ≤ b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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)