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Mirrors > Home > MPE Home > Th. List > r19.27z | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
r19.27z.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
r19.27z | ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 3109 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
2 | r19.27z.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | r19.3rz 4503 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
4 | 3 | anbi2d 630 | . 2 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
5 | 1, 4 | bitr4id 290 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 Ⅎwnf 1780 ≠ wne 2938 ∀wral 3059 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-ne 2939 df-ral 3060 df-dif 3966 df-nul 4340 |
This theorem is referenced by: r19.27zv 4512 raaan 4523 raaan2 4527 |
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