Theorem r19.28z 4521

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)

Hypothesis

Ref	Expression
r19.3rz.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
r19.28z	⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem r19.28z

Step	Hyp	Ref	Expression
1		r19.26 3117	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
2		r19.3rz.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	r19.3rz 4520	. . 3 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
4	3	anbi1d 630	. 2 ⊢ (𝐴 ≠ ∅ → ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
5	1, 4	bitr4id 290	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  Ⅎwnf 1781   ≠ wne 2946  ∀wral 3067  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-ne 2947  df-ral 3068  df-dif 3979  df-nul 4353

This theorem is referenced by:  r19.28zv  4524  raaan  4540  raaan2  4544