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Theorem r19.28v 3181
Description: Restricted quantifier version of one direction of 19.28 2232. (The other direction holds when 𝐴 is nonempty, see r19.28zv 4429.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
Assertion
Ref Expression
r19.28v ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.28v
StepHypRef Expression
1 id 22 . . . 4 (𝜑𝜑)
21ralrimivw 3178 . . 3 (𝜑 → ∀𝑥𝐴 𝜑)
32anim1i 617 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
4 r19.26 3165 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
53, 4sylibr 237 1 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wral 3133
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 1912
This theorem depends on definitions:  df-bi 210  df-an 400  df-ral 3138
This theorem is referenced by:  rr19.28v  3646  fununi  6418  txlm  22251  2reu8i  43535  2reuimp0  43536
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