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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 | r19.27z.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | r19.3rz 4428 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
3 | 2 | anbi2d 630 | . 2 ⊢ (𝐴 ≠ ∅ → ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
4 | r19.26 3170 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
5 | 3, 4 | syl6rbbr 292 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 Ⅎwnf 1784 ≠ wne 3016 ∀wral 3138 ∅c0 4279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-dif 3927 df-nul 4280 |
This theorem is referenced by: r19.27zv 4437 raaan 4446 raaan2 4450 |
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