Theorem li3 252

Description: Introduce Kalmbach implication to the left. (Contributed by NM, 2-Nov-1997.)

Hypothesis

Ref	Expression
li3.1	a = b

Assertion

Ref	Expression
li3	(c →3 a) = (c →3 b)

Proof of Theorem li3

Step	Hyp	Ref	Expression
1		li3.1	. . . . 5 a = b
2	1	lan 77	. . . 4 (c⊥ ∩ a) = (c⊥ ∩ b)
3	1	ax-r4 37	. . . . 5 a⊥ = b⊥
4	3	lan 77	. . . 4 (c⊥ ∩ a⊥ ) = (c⊥ ∩ b⊥ )
5	2, 4	2or 72	. . 3 ((c⊥ ∩ a) ∪ (c⊥ ∩ a⊥ )) = ((c⊥ ∩ b) ∪ (c⊥ ∩ b⊥ ))
6	1	lor 70	. . . 4 (c⊥ ∪ a) = (c⊥ ∪ b)
7	6	lan 77	. . 3 (c ∩ (c⊥ ∪ a)) = (c ∩ (c⊥ ∪ b))
8	5, 7	2or 72	. 2 (((c⊥ ∩ a) ∪ (c⊥ ∩ a⊥ )) ∪ (c ∩ (c⊥ ∪ a))) = (((c⊥ ∩ b) ∪ (c⊥ ∩ b⊥ )) ∪ (c ∩ (c⊥ ∪ b)))
9		df-i3 46	. 2 (c →3 a) = (((c⊥ ∩ a) ∪ (c⊥ ∩ a⊥ )) ∪ (c ∩ (c⊥ ∪ a)))
10		df-i3 46	. 2 (c →3 b) = (((c⊥ ∩ b) ∪ (c⊥ ∩ b⊥ )) ∪ (c ∩ (c⊥ ∪ b)))
11	8, 9, 10	3tr1 63	1 (c →3 a) = (c →3 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This theorem was proved from axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i3 46

This theorem is referenced by:  2i3  254  ud3lem0a  260  bina1  282  i31  520  i3aa  521  i3btr  528  i3li3  539  i3th2  544  i3th3  545  i3th4  546