Theorem ud4lem0b 263

Description: Introduce →₄ to the right. (Contributed by NM, 23-Nov-1997.)

Hypothesis

Ref	Expression
ud4lem0a.1	a = b

Assertion

Ref	Expression
ud4lem0b	(a →4 c) = (b →4 c)

Proof of Theorem ud4lem0b

Step	Hyp	Ref	Expression
1		ud4lem0a.1	. . . . 5 a = b
2	1	ran 78	. . . 4 (a ∩ c) = (b ∩ c)
3	1	ax-r4 37	. . . . 5 a⊥ = b⊥
4	3	ran 78	. . . 4 (a⊥ ∩ c) = (b⊥ ∩ c)
5	2, 4	2or 72	. . 3 ((a ∩ c) ∪ (a⊥ ∩ c)) = ((b ∩ c) ∪ (b⊥ ∩ c))
6	3	ax-r5 38	. . . 4 (a⊥ ∪ c) = (b⊥ ∪ c)
7	6	ran 78	. . 3 ((a⊥ ∪ c) ∩ c⊥ ) = ((b⊥ ∪ c) ∩ c⊥ )
8	5, 7	2or 72	. 2 (((a ∩ c) ∪ (a⊥ ∩ c)) ∪ ((a⊥ ∪ c) ∩ c⊥ )) = (((b ∩ c) ∪ (b⊥ ∩ c)) ∪ ((b⊥ ∪ c) ∩ c⊥ ))
9		df-i4 47	. 2 (a →4 c) = (((a ∩ c) ∪ (a⊥ ∩ c)) ∪ ((a⊥ ∪ c) ∩ c⊥ ))
10		df-i4 47	. 2 (b →4 c) = (((b ∩ c) ∪ (b⊥ ∩ c)) ∪ ((b⊥ ∪ c) ∩ c⊥ ))
11	8, 9, 10	3tr1 63	1 (a →4 c) = (b →4 c)

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 15

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-i4 47

This theorem is used by:  i4i3  271  ud4  598