Theorem u1lem7 754

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u1lem7	(a ->1 (a_|_ ->1 b)) = 1

Proof of Theorem u1lem7

Step	Hyp	Ref	Expression
1		df-i1 43	. 2 (a ->1 (a_|_ ->1 b)) = (a_|_ v (a ^ (a_|_ ->1 b)))
2		ax-a1 29	. . . . . 6 a = a_|__|_
3	2	ran 71	. . . . 5 (a ^ (a_|_ ->1 b)) = (a_|__|_ ^ (a_|_ ->1 b))
4		ancom 68	. . . . . 6 (a_|__|_ ^ (a_|_ ->1 b)) = ((a_|_ ->1 b) ^ a_|__|_)
5		u1lemana 587	. . . . . 6 ((a_|_ ->1 b) ^ a_|__|_) = a_|__|_
6	4, 5	ax-r2 35	. . . . 5 (a_|__|_ ^ (a_|_ ->1 b)) = a_|__|_
7	3, 6	ax-r2 35	. . . 4 (a ^ (a_|_ ->1 b)) = a_|__|_
8	7	lor 66	. . 3 (a_|_ v (a ^ (a_|_ ->1 b))) = (a_|_ v a_|__|_)
9		df-t 40	. . . 4 1 = (a_|_ v a_|__|_)
10	9	ax-r1 34	. . 3 (a_|_ v a_|__|_) = 1
11	8, 10	ax-r2 35	. 2 (a_|_ v (a ^ (a_|_ ->1 b))) = 1
12	1, 11	ax-r2 35	1 (a ->1 (a_|_ ->1 b)) = 1

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  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-i1 43

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