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Metamath Proof Explorer

Theorem 3eltr3i

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017)

Ref Expression
Hypotheses 3eltr3i.1 ⊢ 𝐴 ∈ 𝐵
3eltr3i.2 ⊢ 𝐴 = 𝐶
3eltr3i.3 ⊢ 𝐵 = 𝐷
Assertion 3eltr3i 𝐶 ∈ 𝐷

Proof

Step Hyp Ref Expression
1 3eltr3i.1 ⊢ 𝐴 ∈ 𝐵
2 3eltr3i.2 ⊢ 𝐴 = 𝐶
3 3eltr3i.3 ⊢ 𝐵 = 𝐷
4 1 3 eleqtri ⊢ 𝐴 ∈ 𝐷
5 2 4 eqeltrri ⊢ 𝐶 ∈ 𝐷