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Metamath Proof Explorer

Theorem eleqtrdi

Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006)

Ref Expression
Hypotheses eleqtrdi.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
eleqtrdi.2 ⊢ 𝐵 = 𝐶
Assertion eleqtrdi ( 𝜑 → 𝐴 ∈ 𝐶 )

Proof

Step Hyp Ref Expression
1 eleqtrdi.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2 eleqtrdi.2 ⊢ 𝐵 = 𝐶
3 2 a1i ⊢ ( 𝜑 → 𝐵 = 𝐶 )
4 1 3 eleqtrd ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )