Theorem r19.28v 3188

Description: Restricted quantifier version of one direction of 19.28 2226. (Assuming Ⅎ𝑥𝐴, the other direction holds when 𝐴 is nonempty, see r19.28zv 4507.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)

Assertion

Ref	Expression
r19.28v	⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem r19.28v

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝜑 → 𝜑)
2	1	ralrimivw 3148	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3	2	anim1i 615	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
4		r19.26 3109	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	3, 4	sylibr 234	1 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wral 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908

This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3060

This theorem is referenced by:  rr19.28v  3668  fununi  6643  txlm  23672  2reu8i  47063  2reuimp0  47064