Theorem raleleqOLD

Description: Obsolete version of raleleq as of 9-Mar-2025. (Contributed by AV, 30-Oct-2020) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
2	1	biimpd	⊢ ( 𝐴 = 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
3	2	ralrimiv	⊢ ( 𝐴 = 𝐵 → ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )