Theorem wdf-c1 365

Description: Show that commutator is a 'commutes' analogue for == analogue of =.

Hypothesis

Ref	Expression
wdf-c1.1	(a == ((a ^ b) v (a ^ b_|_))) = 1

Assertion

Ref	Expression
wdf-c1	C (a, b) = 1

Proof of Theorem wdf-c1

Step	Hyp	Ref	Expression
1		cmtrcom 182	. 2 C (a, b) = C (b, a)
2		df-cmtr 126	. 2 C (b, a) = (((b ^ a) v (b ^ a_|_)) v ((b_|_ ^ a) v (b_|_ ^ a_|_)))
3		df-t 40	. . . . 5 1 = (b v b_|_)
4	3	bi1 110	. . . 4 (1 == (b v b_|_)) = 1
5		wdf-c1.1	. . . . . 6 (a == ((a ^ b) v (a ^ b_|_))) = 1
6	5	wcomlem 364	. . . . 5 (b == ((b ^ a) v (b ^ a_|_))) = 1
7		ax-a1 29	. . . . . . . . . . 11 b = b_|__|_
8	7	lan 70	. . . . . . . . . 10 (a ^ b) = (a ^ b_|__|_)
9	8	ax-r5 37	. . . . . . . . 9 ((a ^ b) v (a ^ b_|_)) = ((a ^ b_|__|_) v (a ^ b_|_))
10		ax-a2 30	. . . . . . . . 9 ((a ^ b_|__|_) v (a ^ b_|_)) = ((a ^ b_|_) v (a ^ b_|__|_))
11	9, 10	ax-r2 35	. . . . . . . 8 ((a ^ b) v (a ^ b_|_)) = ((a ^ b_|_) v (a ^ b_|__|_))
12	11	bi1 110	. . . . . . 7 (((a ^ b) v (a ^ b_|_)) == ((a ^ b_|_) v (a ^ b_|__|_))) = 1
13	5, 12	wr2 353	. . . . . 6 (a == ((a ^ b_|_) v (a ^ b_|__|_))) = 1
14	13	wcomlem 364	. . . . 5 (b_|_ == ((b_|_ ^ a) v (b_|_ ^ a_|_))) = 1
15	6, 14	w2or 354	. . . 4 ((b v b_|_) == (((b ^ a) v (b ^ a_|_)) v ((b_|_ ^ a) v (b_|_ ^ a_|_)))) = 1
16	4, 15	wr2 353	. . 3 (1 == (((b ^ a) v (b ^ a_|_)) v ((b_|_ ^ a) v (b_|_ ^ a_|_)))) = 1
17	16	wr3 190	. 2 (((b ^ a) v (b ^ a_|_)) v ((b_|_ ^ a) v (b_|_ ^ a_|_))) = 1
18	1, 2, 17	3tr 62	1 C (a, b) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9   C wcmtr 28

This theorem is referenced by:  wcom0 389  wcom1 390  wlecom 391  wbctr 392  wcbtr 393  wcomcom2 397  wcomcom5 402  wcomdr 403  wcom3i 404  wcom2or 409

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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123  df-cmtr 126

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