Theorem ka4lem 221

Description: Lemma for KA4 soundness (AND version) - uses OL only.

Assertion

Ref	Expression
ka4lem	((a ^ b)_|_ v ((a ^ c) == (b ^ c))) = 1

Proof of Theorem ka4lem

Step	Hyp	Ref	Expression
1		df-a 39	. . . 4 (a ^ b) = (a_|_ v b_|_)_|_
2	1	con2 64	. . 3 (a ^ b)_|_ = (a_|_ v b_|_)
3		df-a 39	. . . . 5 (a ^ c) = (a_|_ v c_|_)_|_
4		df-a 39	. . . . 5 (b ^ c) = (b_|_ v c_|_)_|_
5	3, 4	2bi 91	. . . 4 ((a ^ c) == (b ^ c)) = ((a_|_ v c_|_)_|_ == (b_|_ v c_|_)_|_)
6		conb 114	. . . . 5 ((a_|_ v c_|_) == (b_|_ v c_|_)) = ((a_|_ v c_|_)_|_ == (b_|_ v c_|_)_|_)
7	6	ax-r1 34	. . . 4 ((a_|_ v c_|_)_|_ == (b_|_ v c_|_)_|_) = ((a_|_ v c_|_) == (b_|_ v c_|_))
8	5, 7	ax-r2 35	. . 3 ((a ^ c) == (b ^ c)) = ((a_|_ v c_|_) == (b_|_ v c_|_))
9	2, 8	2or 67	. 2 ((a ^ b)_|_ v ((a ^ c) == (b ^ c))) = ((a_|_ v b_|_) v ((a_|_ v c_|_) == (b_|_ v c_|_)))
10		ka4lemo 220	. 2 ((a_|_ v b_|_) v ((a_|_ v c_|_) == (b_|_ v c_|_))) = 1
11	9, 10	ax-r2 35	1 ((a ^ b)_|_ v ((a ^ c) == (b ^ c))) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

metamath.org