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Theorem r19.27z 4392
Description: Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
Hypothesis
Ref Expression
r19.27z.1 𝑥𝜓
Assertion
Ref Expression
r19.27z (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem r19.27z
StepHypRef Expression
1 r19.26 3085 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
2 r19.27z.1 . . . 4 𝑥𝜓
32r19.3rz 4384 . . 3 (𝐴 ≠ ∅ → (𝜓 ↔ ∀𝑥𝐴 𝜓))
43anbi2d 632 . 2 (𝐴 ≠ ∅ → ((∀𝑥𝐴 𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓)))
51, 4bitr4id 293 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wnf 1790  wne 2935  wral 3054  c0 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-12 2179  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-ne 2936  df-ral 3059  df-dif 3847  df-nul 4213
This theorem is referenced by:  r19.27zv  4393  raaan  4408  raaan2  4412
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