Theorem ri3 253

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

Hypothesis

Ref	Expression
ri3.1	a = b

Assertion

Ref	Expression
ri3	(a →3 c) = (b →3 c)

Proof of Theorem ri3

Step	Hyp	Ref	Expression
1		ri3.1	. . . . . 6 a = b
2	1	ax-r4 37	. . . . 5 a⊥ = b⊥
3	2	ran 78	. . . 4 (a⊥ ∩ c) = (b⊥ ∩ c)
4	2	ran 78	. . . 4 (a⊥ ∩ c⊥ ) = (b⊥ ∩ c⊥ )
5	3, 4	2or 72	. . 3 ((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) = ((b⊥ ∩ c) ∪ (b⊥ ∩ c⊥ ))
6	2	ax-r5 38	. . . 4 (a⊥ ∪ c) = (b⊥ ∪ c)
7	1, 6	2an 79	. . 3 (a ∩ (a⊥ ∪ c)) = (b ∩ (b⊥ ∪ c))
8	5, 7	2or 72	. 2 (((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) ∪ (a ∩ (a⊥ ∪ c))) = (((b⊥ ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ (b ∩ (b⊥ ∪ c)))
9		df-i3 46	. 2 (a →3 c) = (((a⊥ ∩ c) ∪ (a⊥ ∩ c⊥ )) ∪ (a ∩ (a⊥ ∪ c)))
10		df-i3 46	. 2 (b →3 c) = (((b⊥ ∩ c) ∪ (b⊥ ∩ c⊥ )) ∪ (b ∩ (b⊥ ∪ c)))
11	8, 9, 10	3tr1 63	1 (a →3 c) = (b →3 c)

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  ud3lem0b  261  bina2  283  ska14  514  i3orcom  525  i3ancom  526  bi3tr  527  i3ri3  538