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) v (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) v (a' ^ b)) v (a' ^ b'))
7		df-i5 48	. . 3 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
8	7	ax-r1 35	. 2 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (a ->5 b)
9	6, 8	cbtr 182	1 b C (a ->5 b)

Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7   ->5 wi5 16

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

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

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