Theorem i3orlem5 538

Description: Lemma for Kalmbach implication OR builder.

Assertion

Ref	Expression
i3orlem5	((a_|_ ^ b_|_) ^ c_|_) =< ((a v c) ->3 (b v c))

Proof of Theorem i3orlem5

Step	Hyp	Ref	Expression
1		leo 150	. 2 ((a v c)_|_ ^ (b v c)_|_) =< (((a v c)_|_ ^ (b v c)_|_) v (((a v c)_|_ v (b v c)) ^ ((a v c) v ((a v c)_|_ ^ (b v c)))))
2		anandir 107	. . 3 ((a_|_ ^ b_|_) ^ c_|_) = ((a_|_ ^ c_|_) ^ (b_|_ ^ c_|_))
3		oran 79	. . . . . 6 (a v c) = (a_|_ ^ c_|_)_|_
4	3	con2 64	. . . . 5 (a v c)_|_ = (a_|_ ^ c_|_)
5	4	ax-r1 34	. . . 4 (a_|_ ^ c_|_) = (a v c)_|_
6		oran 79	. . . . . 6 (b v c) = (b_|_ ^ c_|_)_|_
7	6	con2 64	. . . . 5 (b v c)_|_ = (b_|_ ^ c_|_)
8	7	ax-r1 34	. . . 4 (b_|_ ^ c_|_) = (b v c)_|_
9	5, 8	2an 72	. . 3 ((a_|_ ^ c_|_) ^ (b_|_ ^ c_|_)) = ((a v c)_|_ ^ (b v c)_|_)
10	2, 9	ax-r2 35	. 2 ((a_|_ ^ b_|_) ^ c_|_) = ((a v c)_|_ ^ (b v c)_|_)
11		df2i3 480	. 2 ((a v c) ->3 (b v c)) = (((a v c)_|_ ^ (b v c)_|_) v (((a v c)_|_ v (b v c)) ^ ((a v c) v ((a v c)_|_ ^ (b v c)))))
12	1, 10, 11	le3tr1 132	1 ((a_|_ ^ b_|_) ^ 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-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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