Theorem ud2lem0b 259

Description: Introduce →₂ to the right. (Contributed by NM, 23-Nov-1997.)

Hypothesis

Ref	Expression
ud2lem0a.1	a = b

Assertion

Ref	Expression
ud2lem0b	(a →2 c) = (b →2 c)

Proof of Theorem ud2lem0b

Step	Hyp	Ref	Expression
1		ud2lem0a.1	. . . . 5 a = b
2	1	ax-r4 37	. . . 4 a⊥ = b⊥
3	2	ran 78	. . 3 (a⊥ ∩ c⊥ ) = (b⊥ ∩ c⊥ )
4	3	lor 70	. 2 (c ∪ (a⊥ ∩ c⊥ )) = (c ∪ (b⊥ ∩ c⊥ ))
5		df-i2 45	. 2 (a →2 c) = (c ∪ (a⊥ ∩ c⊥ ))
6		df-i2 45	. 2 (b →2 c) = (c ∪ (b⊥ ∩ c⊥ ))
7	4, 5, 6	3tr1 63	1 (a →2 c) = (b →2 c)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This proof depends on axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-i2 45

This theorem is used by:  i2i1  267  i1i2con1  268  ud2  596  2oath1i1  827