Theorem raleleq

Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020)

	Ref	Expression
Assertion	raleleq	⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ralel	⊢ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ 𝐵
2		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
3	2	raleqdv	⊢ ( 𝐴 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐵 𝑥 ∈ 𝐵 ) )
4	1 3	mpbiri	⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )