Theorem raleleq

Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020) (Proof shortened by Wolf Lammen, 18-Jul-2025)

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

Proof

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