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| Mirrors > Home > MPE Home > Th. List > 2ralbida | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004.) |
| Ref | Expression |
|---|---|
| 2ralbida.1 | ⊢ Ⅎ𝑥𝜑 |
| 2ralbida.2 | ⊢ Ⅎ𝑦𝜑 |
| 2ralbida.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| 2ralbida | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2ralbida.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | 2ralbida.2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
| 4 | 2, 3 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴) |
| 5 | 2ralbida.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒)) | |
| 6 | 5 | anassrs 467 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝜓 ↔ 𝜒)) |
| 7 | 4, 6 | ralbida 3270 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
| 8 | 1, 7 | ralbida 3270 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 |
| 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-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-ral 3062 |
| This theorem is referenced by: (None) |
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