Theorem i3orlem2 535

Description: Lemma for Kalmbach implication OR builder.

Assertion

Ref	Expression
i3orlem2	(a ^ b) =< ((a v c) ->3 (b v c))

Proof of Theorem i3orlem2

Step	Hyp	Ref	Expression
1		leo 150	. . 3 a =< (a v c)
2		leo 150	. . 3 b =< (b v c)
3	1, 2	le2an 161	. 2 (a ^ b) =< ((a v c) ^ (b v c))
4		leor 151	. . . 4 ((a v c) ^ (b v c)) =< (((a v c) ^ (a v c)_|_) v ((a v c) ^ (b v c)))
5		ledi 166	. . . 4 (((a v c) ^ (a v c)_|_) v ((a v c) ^ (b v c))) =< ((a v c) ^ ((a v c)_|_ v (b v c)))
6	4, 5	letr 129	. . 3 ((a v c) ^ (b v c)) =< ((a v c) ^ ((a v c)_|_ v (b v c)))
7		i3orlem1 534	. . 3 ((a v c) ^ ((a v c)_|_ v (b v c))) =< ((a v c) ->3 (b v c))
8	6, 7	letr 129	. 2 ((a v c) ^ (b v c)) =< ((a v c) ->3 (b v c))
9	3, 8	letr 129	1 (a ^ b) =< ((a v c) ->3 (b v c))

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

This theorem is referenced by:  i3orlem6 539

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

metamath.org