Theorem u2lemc1 681

Description: Commutation theorem for Dishkant implication. (Contributed by NM, 14-Dec-1997.)

Assertion

Ref	Expression
u2lemc1	b C (a →2 b)

Proof of Theorem u2lemc1

Step	Hyp	Ref	Expression
1		comid 187	. . 3 b C b
2		comanr2 465	. . . 4 b⊥ C (a⊥ ∩ b⊥ )
3	2	comcom6 459	. . 3 b C (a⊥ ∩ b⊥ )
4	1, 3	com2or 483	. 2 b C (b ∪ (a⊥ ∩ b⊥ ))
5		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
6	5	ax-r1 35	. 2 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)
7	4, 6	cbtr 182	1 b C (a →2 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u2lemc3  692  u21lembi  727  u2lem3  750  imp3  841  oa23  936