Theorem distid 873

Description: Distributive law for identity.

Assertion

Ref	Expression
distid	((a == b) ^ ((a == c) v (b == c))) = (((a == b) ^ (a == c)) v ((a == b) ^ (b == c)))

Proof of Theorem distid

Step	Hyp	Ref	Expression
1		lea 154	. . . 4 ((a == b) ^ ((a == c) v (b == c))) =< (a == b)
2		mlaconjo 872	. . . 4 ((a == b) ^ ((a == c) v (b == c))) =< (a == c)
3	1, 2	ler2an 167	. . 3 ((a == b) ^ ((a == c) v (b == c))) =< ((a == b) ^ (a == c))
4		bicom 90	. . . . . 6 (a == b) = (b == a)
5		ax-a2 31	. . . . . 6 ((a == c) v (b == c)) = ((b == c) v (a == c))
6	4, 5	2an 73	. . . . 5 ((a == b) ^ ((a == c) v (b == c))) = ((b == a) ^ ((b == c) v (a == c)))
7		mlaconjo 872	. . . . 5 ((b == a) ^ ((b == c) v (a == c))) =< (b == c)
8	6, 7	bltr 132	. . . 4 ((a == b) ^ ((a == c) v (b == c))) =< (b == c)
9	1, 8	ler2an 167	. . 3 ((a == b) ^ ((a == c) v (b == c))) =< ((a == b) ^ (b == c))
10	3, 9	ler2or 166	. 2 ((a == b) ^ ((a == c) v (b == c))) =< (((a == b) ^ (a == c)) v ((a == b) ^ (b == c)))
11		ledi 168	. 2 (((a == b) ^ (a == c)) v ((a == b) ^ (b == c))) =< ((a == b) ^ ((a == c) v (b == c)))
12	10, 11	lebi 139	1 ((a == b) ^ ((a == c) v (b == c))) = (((a == b) ^ (a == c)) v ((a == b) ^ (b == c)))

Syntax hints:   = wb 1   == tb 5   v wo 6   ^ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 425

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 124  df-le2 125  df-c1 126  df-c2 127

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