Theorem li3 252

Description: Introduce Kalmbach implication to the left.

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) v (c' ^ a')) = ((c' ^ b) v (c' ^ b'))
6	1	lor 70	. . . 4 (c' v a) = (c' v b)
7	6	lan 77	. . 3 (c ^ (c' v a)) = (c ^ (c' v b))
8	5, 7	2or 72	. 2 (((c' ^ a) v (c' ^ a')) v (c ^ (c' v a))) = (((c' ^ b) v (c' ^ b')) v (c ^ (c' v b)))
9		df-i3 46	. 2 (c ->3 a) = (((c' ^ a) v (c' ^ a')) v (c ^ (c' v a)))
10		df-i3 46	. 2 (c ->3 b) = (((c' ^ b) v (c' ^ b')) v (c ^ (c' v b)))
11	8, 9, 10	3tr1 63	1 (c ->3 a) = (c ->3 b)

Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14

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

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

Copyright terms: Public domain