Theorem orbile 843

Description: Disjunction of biconditionals. (Contributed by NM, 5-Jul-2000.)

Assertion

Ref	Expression
orbile	((a ≡ c) ∪ (b ≡ c)) ≤ (((a ∩ b) →2 c) ∩ (c →1 (a ∪ b)))

Proof of Theorem orbile

Step	Hyp	Ref	Expression
1		orbi 842	. 2 ((a ≡ c) ∪ (b ≡ c)) = (((a →2 c) ∪ (b →2 c)) ∩ ((c →1 a) ∪ (c →1 b)))
2		i2or 344	. . 3 ((a →2 c) ∪ (b →2 c)) ≤ ((a ∩ b) →2 c)
3		i1or 345	. . 3 ((c →1 a) ∪ (c →1 b)) ≤ (c →1 (a ∪ b))
4	2, 3	le2an 169	. 2 (((a →2 c) ∪ (b →2 c)) ∩ ((c →1 a) ∪ (c →1 b))) ≤ (((a ∩ b) →2 c) ∩ (c →1 (a ∪ b)))
5	1, 4	bltr 138	1 ((a ≡ c) ∪ (b ≡ c)) ≤ (((a ∩ b) →2 c) ∩ (c →1 (a ∪ b)))

Syntax hints:   ≤ wle 2   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →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-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  mlaconj4  844  mlaconj  845  mlaconjolem  885