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Theorem r19.27v 3255
 Description: Restricted quantitifer version of one direction of 19.27 2212. (The other direction holds when 𝐴 is nonempty, see r19.27zv 4293.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
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 ax-1 6 . . . 4 (𝜓 → (𝑥𝐴𝜓))
21ralrimiv 3146 . . 3 (𝜓 → ∀𝑥𝐴 𝜓)
32anim2i 610 . 2 ((∀𝑥𝐴 𝜑𝜓) → (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
4 r19.26 3249 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
53, 4sylibr 226 1 ((∀𝑥𝐴 𝜑𝜓) → ∀𝑥𝐴 (𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2106  ∀wral 3089 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953 This theorem depends on definitions:  df-bi 199  df-an 387  df-ral 3094 This theorem is referenced by:  r19.28v  3256  txlm  21860  tx1stc  21862  spanuni  28975
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