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Theorem r19.27v 3176
Description: Restricted quantitifer version of one direction of 19.27 2230. (The other direction holds when 𝐴 is nonempty, see r19.27zv 4423.) (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 22 . . . 4 (𝜓𝜓)
21ralrimivw 3175 . . 3 (𝜓 → ∀𝑥𝐴 𝜓)
32anim2i 619 . 2 ((∀𝑥𝐴 𝜑𝜓) → (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
4 r19.26 3162 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
53, 4sylibr 237 1 ((∀𝑥𝐴 𝜑𝜓) → ∀𝑥𝐴 (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wral 3130
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 3135
This theorem is referenced by:  txlm  22251  tx1stc  22253  spanuni  29325
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