Theorem r19.27z 4505

Description: Restricted quantifier version of Theorem 19.27 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
r19.27z.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
r19.27z	⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem r19.27z

Step	Hyp	Ref	Expression
1		r19.26 3111	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
2		r19.27z.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	r19.3rz 4497	. . 3 ⊢ (𝐴 ≠ ∅ → (𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜓))
4	3	anbi2d 630	. 2 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
5	1, 4	bitr4id 290	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  Ⅎwnf 1783   ≠ wne 2940  ∀wral 3061  ∅c0 4333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-ne 2941  df-ral 3062  df-dif 3954  df-nul 4334

This theorem is referenced by:  r19.27zv  4506  raaan  4517  raaan2  4521