Theorem negantlem4 851

Description: Lemma for negated antecedent identity. (Contributed by NM, 6-Aug-2001.)

Hypothesis

Ref	Expression
negant.1	(a →1 c) = (b →1 c)

Assertion

Ref	Expression
negantlem4	(a⊥ →1 c) ≤ (b⊥ →1 c)

Proof of Theorem negantlem4

Step	Hyp	Ref	Expression
1		df-i1 44	. . 3 (a⊥ →1 c) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
2		ax-a1 30	. . . . 5 a = a⊥ ⊥
3	2	ax-r5 38	. . . 4 (a ∪ (a⊥ ∩ c)) = (a⊥ ⊥ ∪ (a⊥ ∩ c))
4	3	ax-r1 35	. . 3 (a⊥ ⊥ ∪ (a⊥ ∩ c)) = (a ∪ (a⊥ ∩ c))
5	1, 4	ax-r2 36	. 2 (a⊥ →1 c) = (a ∪ (a⊥ ∩ c))
6		negant.1	. . . 4 (a →1 c) = (b →1 c)
7	6	negantlem2 849	. . 3 a ≤ (b⊥ →1 c)
8	6	negantlem3 850	. . 3 (a⊥ ∩ c) ≤ (b⊥ →1 c)
9	7, 8	lel2or 170	. 2 (a ∪ (a⊥ ∩ c)) ≤ (b⊥ →1 c)
10	5, 9	bltr 138	1 (a⊥ →1 c) ≤ (b⊥ →1 c)

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:  negant  852