Theorem u5lemc2 672

Description: Commutation theorem for relevance implication.

Hypotheses

Ref	Expression
ulemc2.1	a C b
ulemc2.2	a C c

Assertion

Ref	Expression
u5lemc2	a C (b →5 c)

Proof of Theorem u5lemc2

Step	Hyp	Ref	Expression
1		ulemc2.1	. . . . 5 a C b
2		ulemc2.2	. . . . 5 a C c
3	1, 2	com2an 466	. . . 4 a C (b ∩ c)
4	1	comcom2 175	. . . . 5 a C b⊥
5	4, 2	com2an 466	. . . 4 a C (b⊥ ∩ c)
6	3, 5	com2or 465	. . 3 a C ((b ∩ c) ∪ (b⊥ ∩ c))
7	2	comcom2 175	. . . 4 a C c⊥
8	4, 7	com2an 466	. . 3 a C (b⊥ ∩ c⊥ )
9	6, 8	com2or 465	. 2 a C (((b ∩ c) ∪ (b⊥ ∩ c)) ∪ (b⊥ ∩ c⊥ ))
10		df-i5 47	. . 3 (b →5 c) = (((b ∩ c) ∪ (b⊥ ∩ c)) ∪ (b⊥ ∩ c⊥ ))
11	10	ax-r1 34	. 2 (((b ∩ c) ∪ (b⊥ ∩ c)) ∪ (b⊥ ∩ c⊥ )) = (b →5 c)
12	9, 11	cbtr 174	1 a C (b →5 c)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 17

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-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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