Theorem eqeltr

Description: Substitution of equal classes into elementhood relation. (Contributed by Peter Mazsa, 22-Jul-2017)

	Ref	Expression
Assertion	eqeltr	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
2	1	biimpar	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 )