Theorem i3orlem3 536

Description: Lemma for Kalmbach implication OR builder.

Assertion

Ref	Expression
i3orlem3	c =< ((a v c) ->3 (b v c))

Proof of Theorem i3orlem3

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . . . 6 ((a v c)_|_ v c) = (c v (a v c)_|_)
2	1	lan 70	. . . . 5 (c ^ ((a v c)_|_ v c)) = (c ^ (c v (a v c)_|_))
3		a5c 113	. . . . 5 (c ^ (c v (a v c)_|_)) = c
4	2, 3	ax-r2 35	. . . 4 (c ^ ((a v c)_|_ v c)) = c
5	4	ax-r1 34	. . 3 c = (c ^ ((a v c)_|_ v c))
6		leor 151	. . . 4 c =< (a v c)
7		leor 151	. . . . 5 c =< (b v c)
8	7	lelor 158	. . . 4 ((a v c)_|_ v c) =< ((a v c)_|_ v (b v c))
9	6, 8	le2an 161	. . 3 (c ^ ((a v c)_|_ v c)) =< ((a v c) ^ ((a v c)_|_ v (b v c)))
10	5, 9	bltr 130	. 2 c =< ((a v c) ^ ((a v c)_|_ v (b v c)))
11		i3orlem1 534	. 2 ((a v c) ^ ((a v c)_|_ v (b v c))) =< ((a v c) ->3 (b v c))
12	10, 11	letr 129	1 c =< ((a v c) ->3 (b v c))

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->3 wi3 15

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123

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