Theorem r19.27v 3151

Description: Restricted quantitifer version of one direction of 19.27 2227. (The other direction holds when 𝐴 is nonempty, see r19.27zv 4409.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)

Assertion

Ref	Expression
r19.27v	⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem r19.27v

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝜓 → 𝜓)
2	1	ralrimivw 3150	. . 3 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜓)
3	2	anim2i 619	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
4		r19.26 3137	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
5	3, 4	sylibr 237	1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 399  ∀wral 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911

This theorem depends on definitions:  df-bi 210  df-an 400  df-ral 3111

This theorem is referenced by:  txlm  22253  tx1stc  22255  spanuni  29327