This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ertrd

Description: A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014)

Ref Expression
Hypotheses ersymb.1 ⊢ φ → R Er X
ertrd.5 ⊢ φ → A R B
ertrd.6 ⊢ φ → B R C
Assertion ertrd ⊢ φ → A R C

Proof

Step Hyp Ref Expression
1 ersymb.1 ⊢ φ → R Er X
2 ertrd.5 ⊢ φ → A R B
3 ertrd.6 ⊢ φ → B R C
4 1 ertr ⊢ φ → A R B ∧ B R C → A R C
5 2 3 4 mp2and ⊢ φ → A R C