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Metamath Proof Explorer

Theorem 3eltr4g

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypotheses	3eltr4g.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	3eltr4g.2	⊢ 𝐶 = 𝐴
	3eltr4g.3	⊢ 𝐷 = 𝐵
Assertion	3eltr4g	⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3eltr4g.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		3eltr4g.2	⊢ 𝐶 = 𝐴
3		3eltr4g.3	⊢ 𝐷 = 𝐵
4	2 1	eqeltrid	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )
5	4 3	eleqtrrdi	⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )