Theorem r19.21bi

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994) (Proof shortened by Wolf Lammen, 11-Jun-2023)

	Ref	Expression
Hypothesis	r19.21bi.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
Assertion	r19.21bi	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		r19.21bi.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
2		rspa	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜓 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 )
3	1 2	sylan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝜓 )