Theorem r19.36zv 4434

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 3268	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
2		r19.9rzv 4427	. . 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 289	1 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-ne 2943  df-ral 3068  df-rex 3069  df-dif 3886  df-nul 4254

This theorem is referenced by:  2reuimp  44494