Theorem u2lemc2 669

Description: Commutation theorem for Dishkant implication.

Hypotheses

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

Assertion

Ref	Expression
u2lemc2	a C (b →2 c)

Proof of Theorem u2lemc2

Step	Hyp	Ref	Expression
1		ulemc2.2	. . 3 a C c
2		ulemc2.1	. . . . 5 a C b
3	2	comcom2 175	. . . 4 a C b⊥
4	1	comcom2 175	. . . 4 a C c⊥
5	3, 4	com2an 466	. . 3 a C (b⊥ ∩ c⊥ )
6	1, 5	com2or 465	. 2 a C (c ∪ (b⊥ ∩ c⊥ ))
7		df-i2 44	. . 3 (b →2 c) = (c ∪ (b⊥ ∩ c⊥ ))
8	7	ax-r1 34	. 2 (c ∪ (b⊥ ∩ c⊥ )) = (b →2 c)
9	6, 8	cbtr 174	1 a C (b →2 c)

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

This theorem is referenced by:  u2lemc5 679

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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