Theorem wbctr 410

Description: Transitive inference. (Contributed by NM, 13-Oct-1997.)

Hypotheses

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

Assertion

Ref	Expression
wbctr	C (a, c) = 1

Proof of Theorem wbctr

Step	Hyp	Ref	Expression
1		wbctr.2	. . . 4 C (b, c) = 1
2	1	wdf-c2 384	. . 3 (b ≡ ((b ∩ c) ∪ (b ∩ c⊥ ))) = 1
3		wbctr.1	. . 3 (a ≡ b) = 1
4	3	wran 369	. . . 4 ((a ∩ c) ≡ (b ∩ c)) = 1
5	3	wran 369	. . . 4 ((a ∩ c⊥ ) ≡ (b ∩ c⊥ )) = 1
6	4, 5	w2or 372	. . 3 (((a ∩ c) ∪ (a ∩ c⊥ )) ≡ ((b ∩ c) ∪ (b ∩ c⊥ ))) = 1
7	2, 3, 6	w3tr1 374	. 2 (a ≡ ((a ∩ c) ∪ (a ∩ c⊥ ))) = 1
8	7	wdf-c1 383	1 C (a, c) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by:  woml7  437