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Metamath Proof Explorer

Theorem r19.36zv

Description: Restricted quantifier version of Theorem 19.36 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003)

		Ref	Expression
	Assertion	r19.36zv	⊢ ( 𝐴 ≠ ∅ → ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		r19.35	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
2		r19.9rzv	⊢ ( 𝐴 ≠ ∅ → ( 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 𝜓 ) )
3	2	imbi2d	⊢ ( 𝐴 ≠ ∅ → ( ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) ) )
4	1 3	bitr4id	⊢ ( 𝐴 ≠ ∅ → ( ∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝜓 ) ) )