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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > r19.28zf | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
r19.28zf.1 | ⊢ Ⅎ𝑥𝜑 |
r19.28zf.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
r19.28zf | ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3112 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
2 | r19.28zf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | r19.28zf.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | r19.3rzf 43785 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
5 | 4 | anbi1d 631 | . 2 ⊢ (𝐴 ≠ ∅ → ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
6 | 1, 5 | bitr4id 290 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 Ⅎwnf 1786 Ⅎwnfc 2884 ≠ wne 2941 ∀wral 3062 ∅c0 4321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-dif 3950 df-nul 4322 |
This theorem is referenced by: iindif2f 43787 |
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