Theorem r19.36zv 4464

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.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem r19.36zv

Step	Hyp	Ref	Expression
1		r19.35 3093	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		r19.9rzv 4457	. . 3 ⊢ (𝐴 ≠ ∅ → (𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓))
3	2	imbi2d 340	. 2 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)))
4	1, 3	bitr4id 290	1 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  ∅c0 4284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-ne 2932  df-ral 3051  df-rex 3060  df-dif 3903  df-nul 4285

This theorem is referenced by:  2reuimp  47398