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Metamath Proof Explorer

Theorem ertr4d

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

Ref Expression
Hypotheses ersymb.1 ⊢ φ → R Er X
ertr4d.5 ⊢ φ → A R B
ertr4d.6 ⊢ φ → C R B
Assertion ertr4d ⊢ φ → A R C

Proof

Step Hyp Ref Expression
1 ersymb.1 ⊢ φ → R Er X
2 ertr4d.5 ⊢ φ → A R B
3 ertr4d.6 ⊢ φ → C R B
4 1 3 ersym ⊢ φ → B R C
5 1 2 4 ertrd ⊢ φ → A R C