Theorem eqab2

Description: Implication of a class abstraction. (Contributed by Peter Mazsa, 16-Apr-2019)

	Ref	Expression
Assertion	eqab2	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) → ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		biimp	⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
2	1	alimi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
4	2 3	sylibr	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) → ∀ 𝑥 ∈ 𝐴 𝜑 )