Theorem wcomcom2 397

Description: Commutation equivalence. Kalmbach 83 p. 23.

Hypothesis

Ref	Expression
wcomcom.1	C (a, b) = 1

Assertion

Ref	Expression
wcomcom2	C (a, b_|_) = 1

Proof of Theorem wcomcom2

Step	Hyp	Ref	Expression
1		wcomcom.1	. . . . 5 C (a, b) = 1
2	1	wdf-c2 366	. . . 4 (a == ((a ^ b) v (a ^ b_|_))) = 1
3		ax-a1 29	. . . . . . 7 b = b_|__|_
4	3	bi1 110	. . . . . 6 (b == b_|__|_) = 1
5	4	wlan 352	. . . . 5 ((a ^ b) == (a ^ b_|__|_)) = 1
6	5	wr5-2v 348	. . . 4 (((a ^ b) v (a ^ b_|_)) == ((a ^ b_|__|_) v (a ^ b_|_))) = 1
7	2, 6	wr2 353	. . 3 (a == ((a ^ b_|__|_) v (a ^ b_|_))) = 1
8		ax-a2 30	. . . 4 ((a ^ b_|__|_) v (a ^ b_|_)) = ((a ^ b_|_) v (a ^ b_|__|_))
9	8	bi1 110	. . 3 (((a ^ b_|__|_) v (a ^ b_|_)) == ((a ^ b_|_) v (a ^ b_|__|_))) = 1
10	7, 9	wr2 353	. 2 (a == ((a ^ b_|_) v (a ^ b_|__|_))) = 1
11	10	wdf-c1 365	1 C (a, b_|_) = 1

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

This theorem is referenced by:  wcomcom3 398  wcomcom4 399  wfh1 405  wfh2 406  wnbdi 411  ska2 414  ska4 415

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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