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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 3095 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
2 | r19.27z.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | r19.3rz 4427 | . . 3 ⊢ (𝐴 ≠ ∅ → (𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜓)) |
4 | 3 | anbi2d 629 | . 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 205 ∧ wa 396 Ⅎwnf 1786 ≠ wne 2943 ∀wral 3064 ∅c0 4256 |
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 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-ne 2944 df-ral 3069 df-dif 3890 df-nul 4257 |
This theorem is referenced by: r19.27zv 4436 raaan 4451 raaan2 4455 |
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