Theorem u2lemc1 663

Description: Commutation theorem for Dishkant implication.

Assertion

Ref	Expression
u2lemc1	b C (a →2 b)

Proof of Theorem u2lemc1

Step	Hyp	Ref	Expression
1		comid 179	. . 3 b C b
2		comanr2 447	. . . 4 b⊥ C (a⊥ ∩ b⊥ )
3	2	comcom6 441	. . 3 b C (a⊥ ∩ b⊥ )
4	1, 3	com2or 465	. 2 b C (b ∪ (a⊥ ∩ b⊥ ))
5		df-i2 44	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
6	5	ax-r1 34	. 2 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)
7	4, 6	cbtr 174	1 b C (a →2 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 14

This theorem is referenced by:  u2lemc3 674  u21lembi 709  u2lem3 732  imp3 823  oa23 916

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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