Theorem eqtr4 834

Description: 4-variable transitive law for equivalence. (Contributed by NM, 26-Jun-2003.)

Assertion

Ref	Expression
eqtr4	(((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ d)) ≤ (a ≡ d)

Proof of Theorem eqtr4

Step	Hyp	Ref	Expression
1		mlaoml 833	. . 3 ((a ≡ b) ∩ (b ≡ c)) ≤ (a ≡ c)
2	1	leran 153	. 2 (((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ d)) ≤ ((a ≡ c) ∩ (c ≡ d))
3		mlaoml 833	. 2 ((a ≡ c) ∩ (c ≡ d)) ≤ (a ≡ d)
4	2, 3	letr 137	1 (((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ d)) ≤ (a ≡ d)

Syntax hints:   ≤ wle 2   ≡ tb 5   ∩ wa 7

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-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  oago3.21x  890