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Theorem r19.28zf 45066
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
Hypotheses
Ref Expression
r19.28zf.1 𝑥𝜑
r19.28zf.2 𝑥𝐴
Assertion
Ref Expression
r19.28zf (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Proof of Theorem r19.28zf
StepHypRef Expression
1 r19.26 3117 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
2 r19.28zf.1 . . . 4 𝑥𝜑
3 r19.28zf.2 . . . 4 𝑥𝐴
42, 3r19.3rzf 45065 . . 3 (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))
54anbi1d 630 . 2 (𝐴 ≠ ∅ → ((𝜑 ∧ ∀𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓)))
61, 5bitr4id 290 1 (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wnf 1781  wnfc 2893  wne 2946  wral 3067  c0 4352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-dif 3979  df-nul 4353
This theorem is referenced by:  iindif2f  45067
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