Theorem distid 869

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 152	. . . 4 ((a == b) ^ ((a == c) v (b == c))) =< (a == b)
2		mlaconjo 868	. . . 4 ((a == b) ^ ((a == c) v (b == c))) =< (a == c)
3	1, 2	ler2an 165	. . 3 ((a == b) ^ ((a == c) v (b == c))) =< ((a == b) ^ (a == c))
4		bicom 88	. . . . . 6 (a == b) = (b == a)
5		ax-a2 30	. . . . . 6 ((a == c) v (b == c)) = ((b == c) v (a == c))
6	4, 5	2an 72	. . . . 5 ((a == b) ^ ((a == c) v (b == c))) = ((b == a) ^ ((b == c) v (a == c)))
7		mlaconjo 868	. . . . 5 ((b == a) ^ ((b == c) v (a == c))) =< (b == c)
8	6, 7	bltr 130	. . . 4 ((a == b) ^ ((a == c) v (b == c))) =< (b == c)
9	1, 8	ler2an 165	. . 3 ((a == b) ^ ((a == c) v (b == c))) =< ((a == b) ^ (b == c))
10	3, 9	ler2or 164	. 2 ((a == b) ^ ((a == c) v (b == c))) =< (((a == b) ^ (a == c)) v ((a == b) ^ (b == c)))
11		ledi 166	. 2 (((a == b) ^ (a == c)) v ((a == b) ^ (b == c))) =< ((a == b) ^ ((a == c) v (b == c)))
12	10, 11	lebi 137	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 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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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