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Metamath Proof Explorer

Theorem 3eltr4d

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

Ref Expression
Hypotheses 3eltr4d.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
3eltr4d.2 ⊢ ( 𝜑 → 𝐶 = 𝐴 )
3eltr4d.3 ⊢ ( 𝜑 → 𝐷 = 𝐵 )
Assertion 3eltr4d ( 𝜑 → 𝐶 ∈ 𝐷 )

Proof

Step Hyp Ref Expression
1 3eltr4d.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2 3eltr4d.2 ⊢ ( 𝜑 → 𝐶 = 𝐴 )
3 3eltr4d.3 ⊢ ( 𝜑 → 𝐷 = 𝐵 )
4 1 3 eleqtrrd ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
5 2 4 eqeltrd ⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )