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Metamath Proof Explorer

Theorem 3eltr3d

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

Ref Expression
Hypotheses 3eltr3d.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
3eltr3d.2 ⊢ ( 𝜑 → 𝐴 = 𝐶 )
3eltr3d.3 ⊢ ( 𝜑 → 𝐵 = 𝐷 )
Assertion 3eltr3d ( 𝜑 → 𝐶 ∈ 𝐷 )

Proof

Step Hyp Ref Expression
1 3eltr3d.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2 3eltr3d.2 ⊢ ( 𝜑 → 𝐴 = 𝐶 )
3 3eltr3d.3 ⊢ ( 𝜑 → 𝐵 = 𝐷 )
4 1 3 eleqtrd ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
5 2 4 eqeltrrd ⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )