Theorem eqtr4 816

Description: 4-variable transitive law for equivalence.

Assertion

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

Proof of Theorem eqtr4

Step	Hyp	Ref	Expression
1		mlaoml 815	. . 3 ((a ≡ b) ∩ (b ≡ c)) ≤ (a ≡ c)
2	1	leran 145	. 2 (((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ d)) ≤ ((a ≡ c) ∩ (c ≡ d))
3		mlaoml 815	. 2 ((a ≡ c) ∩ (c ≡ d)) ≤ (a ≡ d)
4	2, 3	letr 129	1 (((a ≡ b) ∩ (b ≡ c)) ∩ (c ≡ d)) ≤ (a ≡ d)

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

This theorem is referenced by:  oago3.21x 872

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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