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Theorem r19.27v 2584
 Description: Restricted quantitifer version of one direction of 19.27 1541. (The other direction holds when 𝐴 is inhabited, see r19.27mv 3490.) (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 ((∀𝑥𝐴 𝜑𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem r19.27v
StepHypRef Expression
1 id 19 . . . 4 (𝜓𝜓)
21ralrimivw 2531 . . 3 (𝜓 → ∀𝑥𝐴 𝜓)
32anim2i 340 . 2 ((∀𝑥𝐴 𝜑𝜓) → (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
4 r19.26 2583 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
53, 4sylibr 133 1 ((∀𝑥𝐴 𝜑𝜓) → ∀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  12639
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