Theorem bnj1459

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1459.1	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) )
	bnj1459.2	⊢ ( 𝜓 → 𝜒 )
Assertion	bnj1459	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		bnj1459.1	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) )
2		bnj1459.2	⊢ ( 𝜓 → 𝜒 )
3	1 2	sylbir	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜒 )
4	3	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜒 )