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Metamath Proof Explorer

Theorem 3eltr4i

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

Ref Expression
Hypotheses 3eltr4i.1 ⊢ 𝐴 ∈ 𝐵
3eltr4i.2 ⊢ 𝐶 = 𝐴
3eltr4i.3 ⊢ 𝐷 = 𝐵
Assertion 3eltr4i 𝐶 ∈ 𝐷

Proof

Step Hyp Ref Expression
1 3eltr4i.1 ⊢ 𝐴 ∈ 𝐵
2 3eltr4i.2 ⊢ 𝐶 = 𝐴
3 3eltr4i.3 ⊢ 𝐷 = 𝐵
4 1 3 eleqtrri ⊢ 𝐴 ∈ 𝐷
5 2 4 eqeltri ⊢ 𝐶 ∈ 𝐷