Theorem oagen2 996

Description: "Generalized" OA.

Hypothesis

Ref	Expression
oagen2.1	d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))

Assertion

Ref	Expression
oagen2	((a ->2 b) ^ d) =< (a ->2 c)

Proof of Theorem oagen2

Step	Hyp	Ref	Expression
1		oagen2.1	. . . 4 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2		df-i0 42	. . . 4 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
3	1, 2	lbtr 131	. . 3 d =< ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
4	3	lelan 159	. 2 ((a ->2 b) ^ d) =< ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
5		oal2 979	. 2 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
6	4, 5	letr 129	1 ((a ->2 b) ^ d) =< (a ->2 c)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->0 wi0 12   ->2 wi2 14

This theorem is referenced by:  oagen2b 997

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  ax-3oa 978

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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