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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 3111 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
| 2 | r19.27z.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 3 | 2 | r19.3rz 4497 | . . 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 1783 ≠ wne 2940 ∀wral 3061 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-ne 2941 df-ral 3062 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: r19.27zv 4506 raaan 4517 raaan2 4521 | 
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