Theorem oaeqv 812

Description: Weakened OA implies OA).

Hypothesis

Ref	Expression
oaeqv.1	((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))

Assertion

Ref	Expression
oaeqv	((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)

Proof of Theorem oaeqv

Step	Hyp	Ref	Expression
1		lea 152	. . . 4 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 b)
2		oaeqv.1	. . . 4 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))
3	1, 2	ler2an 165	. . 3 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
4		2oath1 808	. . 3 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
5	3, 4	lbtr 131	. 2 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ (a ->2 c))
6		lear 153	. 2 ((a ->2 b) ^ (a ->2 c)) =< (a ->2 c)
7	5, 6	letr 129	1 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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