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Metamath Proof Explorer

Theorem eqeltrrd

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004)

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

Proof

Step Hyp Ref Expression
1 eqeltrrd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 eqeltrrd.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
3 1 eqcomd ⊢ ( 𝜑 → 𝐵 = 𝐴 )
4 3 2 eqeltrd ⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )