Theorem comanb 854

Description: Biconditional commutation law.

Assertion

Ref	Expression
comanb	(a ^ b) C ((a == c) ^ (b == c))

Proof of Theorem comanb

Step	Hyp	Ref	Expression
1		lea 152	. . . 4 (((a v c)_|_ v ((a ^ b) ^ c)) ^ (b ->1 c)) =< ((a v c)_|_ v ((a ^ b) ^ c))
2		lea 152	. . . . . . 7 (a ^ b) =< a
3		leo 150	. . . . . . 7 a =< (a v c)
4	2, 3	letr 129	. . . . . 6 (a ^ b) =< (a v c)
5	4	lecon 146	. . . . 5 (a v c)_|_ =< (a ^ b)_|_
6	5	leror 144	. . . 4 ((a v c)_|_ v ((a ^ b) ^ c)) =< ((a ^ b)_|_ v ((a ^ b) ^ c))
7	1, 6	letr 129	. . 3 (((a v c)_|_ v ((a ^ b) ^ c)) ^ (b ->1 c)) =< ((a ^ b)_|_ v ((a ^ b) ^ c))
8		comanblem1 852	. . 3 ((a == c) ^ (b == c)) = (((a v c)_|_ v ((a ^ b) ^ c)) ^ (b ->1 c))
9		df-i1 43	. . . 4 ((a ^ b) ->1 ((a == c) ^ (b == c))) = ((a ^ b)_|_ v ((a ^ b) ^ ((a == c) ^ (b == c))))
10		comanblem2 853	. . . . 5 ((a ^ b) ^ ((a == c) ^ (b == c))) = ((a ^ b) ^ c)
11	10	lor 66	. . . 4 ((a ^ b)_|_ v ((a ^ b) ^ ((a == c) ^ (b == c)))) = ((a ^ b)_|_ v ((a ^ b) ^ c))
12	9, 11	ax-r2 35	. . 3 ((a ^ b) ->1 ((a == c) ^ (b == c))) = ((a ^ b)_|_ v ((a ^ b) ^ c))
13	7, 8, 12	le3tr1 132	. 2 ((a == c) ^ (b == c)) =< ((a ^ b) ->1 ((a == c) ^ (b == c)))
14	13	i1com 690	1 (a ^ b) C ((a == c) ^ (b == c))

Syntax hints:   C wc 3  _|_wn 4   == tb 5   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  comanbn 855

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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