Theorem u3lem11a 769

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u3lem11a	(a ->3 ((b ->3 a) ->3 (a ->3 b))_|_) = (a ->3 b_|_)

Proof of Theorem u3lem11a

Step	Hyp	Ref	Expression
1		ud3lem1 552	. . . . 5 ((b ->3 a) ->3 (a ->3 b)) = (b v (b_|_ ^ a_|_))
2		ancom 68	. . . . . . . 8 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
3		anor3 82	. . . . . . . 8 (a_|_ ^ b_|_) = (a v b)_|_
4	2, 3	ax-r2 35	. . . . . . 7 (b_|_ ^ a_|_) = (a v b)_|_
5	4	lor 66	. . . . . 6 (b v (b_|_ ^ a_|_)) = (b v (a v b)_|_)
6		oran1 83	. . . . . 6 (b v (a v b)_|_) = (b_|_ ^ (a v b))_|_
7	5, 6	ax-r2 35	. . . . 5 (b v (b_|_ ^ a_|_)) = (b_|_ ^ (a v b))_|_
8	1, 7	ax-r2 35	. . . 4 ((b ->3 a) ->3 (a ->3 b)) = (b_|_ ^ (a v b))_|_
9	8	con2 64	. . 3 ((b ->3 a) ->3 (a ->3 b))_|_ = (b_|_ ^ (a v b))
10	9	ud3lem0a 252	. 2 (a ->3 ((b ->3 a) ->3 (a ->3 b))_|_) = (a ->3 (b_|_ ^ (a v b)))
11		u3lem11 768	. 2 (a ->3 (b_|_ ^ (a v b))) = (a ->3 b_|_)
12	10, 11	ax-r2 35	1 (a ->3 ((b ->3 a) ->3 (a ->3 b))_|_) = (a ->3 b_|_)

Syntax hints:   = wb 1  _|_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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