Theorem wcbtr 393

Description: Transitive inference.

Hypotheses

Ref	Expression
wcbtr.1	C (a, b) = 1
wcbtr.2	(b ≡ c) = 1

Assertion

Ref	Expression
wcbtr	C (a, c) = 1

Proof of Theorem wcbtr

Step	Hyp	Ref	Expression
1		wcbtr.1	. . . 4 C (a, b) = 1
2	1	wdf-c2 366	. . 3 (a ≡ ((a ∩ b) ∪ (a ∩ b⊥ ))) = 1
3		wcbtr.2	. . . . 5 (b ≡ c) = 1
4	3	wlan 352	. . . 4 ((a ∩ b) ≡ (a ∩ c)) = 1
5	3	wr4 191	. . . . 5 (b⊥ ≡ c⊥ ) = 1
6	5	wlan 352	. . . 4 ((a ∩ b⊥ ) ≡ (a ∩ c⊥ )) = 1
7	4, 6	w2or 354	. . 3 (((a ∩ b) ∪ (a ∩ b⊥ )) ≡ ((a ∩ c) ∪ (a ∩ c⊥ ))) = 1
8	2, 7	wr2 353	. 2 (a ≡ ((a ∩ c) ∪ (a ∩ c⊥ ))) = 1
9	8	wdf-c1 365	1 C (a, c) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9   C wcmtr 28

This theorem is referenced by:  wcom2an 410  wnbdi 411  ska2 414  ska4 415  woml6 418

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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