Theorem r19.28v 3195

Description: Restricted quantifier version of one direction of 19.28 2263. (Assuming Ⅎ𝑥𝐴, the other direction holds when 𝐴 is nonempty, see r19.28zv 4466.) (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 23	. . . 4 ⊢ (𝜑 → 𝜑)
2	1	ralrimivw 3160	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
3	2	anim1i 626	. 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
4		r19.26 3124	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	3, 4	sylibr 237	1 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∀wral 3078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939

This proof depends on definitions:  df-bi 210  df-an 401  df-ral 3079

This theorem is used by:  rr19.28v  3626  fununi  6611  txlm  23816  2reu8i  47878  2reuimp0  47879