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Theorem r19.28v 2585
 Description: Restricted quantifier version of one direction of 19.28 1543. (The other direction holds when 𝐴 is inhabited, see r19.28mv 3486.) (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 19 . . . 4 (𝜑𝜑)
21ralrimivw 2531 . . 3 (𝜑 → ∀𝑥𝐴 𝜑)
32anim1i 338 . 2 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
4 r19.26 2583 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
53, 4sylibr 133 1 ((𝜑 ∧ ∀𝑥𝐴 𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wral 2435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-4 1490  ax-17 1506 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440 This theorem is referenced by:  txlm  12650
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