Theorem u5lemc1b 685

Description: Commutation theorem for relevance implication.

Assertion

Ref	Expression
u5lemc1b	b C (a →5 b)

Proof of Theorem u5lemc1b

Step	Hyp	Ref	Expression
1		comanr2 465	. . . 4 b C (a ∩ b)
2		comanr2 465	. . . 4 b C (a⊥ ∩ b)
3	1, 2	com2or 483	. . 3 b C ((a ∩ b) ∪ (a⊥ ∩ b))
4		comanr2 465	. . . 4 b⊥ C (a⊥ ∩ b⊥ )
5	4	comcom6 459	. . 3 b C (a⊥ ∩ b⊥ )
6	3, 5	com2or 483	. 2 b C (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
7		df-i5 48	. . 3 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
8	7	ax-r1 35	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = (a →5 b)
9	6, 8	cbtr 182	1 b C (a →5 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 16

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-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u5lemc3  695  u5lembi  725  u5lem2  748  u5lem3  753