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| Mirrors > Home > MPE Home > Th. List > aaanOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of aaan 2333 as of 21-Nov-2024. (Contributed by NM, 12-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| aaan.1 | ⊢ Ⅎ𝑦𝜑 | 
| aaan.2 | ⊢ Ⅎ𝑥𝜓 | 
| Ref | Expression | 
|---|---|
| aaanOLD | ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | aaan.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 2 | 1 | 19.28 2228 | . . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓)) | 
| 3 | 2 | albii 1819 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓)) | 
| 4 | aaan.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 5 | 4 | nfal 2323 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 | 
| 6 | 5 | 19.27 2227 | . 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) | 
| 7 | 3, 6 | bitri 275 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∀wal 1538 Ⅎwnf 1783 | 
| 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-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: (None) | 
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