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Metamath Proof Explorer

Theorem bnj228

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (Proof shortened by Andrew Salmon, 9-Jul-2011) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	bnj228.1	⊢ ( 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 )
	Assertion	bnj228	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bnj228.1	⊢ ( 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 )
2		rsp	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 → ( 𝑥 ∈ 𝐴 → 𝜓 ) )
3	1 2	sylbi	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜓 ) )
4	3	impcom	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 )