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| Mirrors > Home > MPE Home > Th. List > 19.12vv | Structured version Visualization version GIF version | ||
| Description: Special case of 19.12 2327 where its converse holds. See 19.12vvv 1988 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.) |
| Ref | Expression |
|---|---|
| 19.12vv | ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 19.21v 1939 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)) | |
| 2 | 1 | exbii 1848 | . 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓)) |
| 3 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
| 4 | 3 | nfal 2323 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 |
| 5 | 4 | 19.36 2230 | . 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓)) |
| 6 | 19.36v 1987 | . . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) | |
| 7 | 6 | albii 1819 | . . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓)) |
| 8 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 9 | 8 | nfal 2323 | . . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑 |
| 10 | 9 | 19.21 2207 | . . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓)) |
| 11 | 7, 10 | bitr2i 276 | . 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) |
| 12 | 2, 5, 11 | 3bitri 297 | 1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 |
| 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-ex 1780 df-nf 1784 |
| This theorem is referenced by: (None) |
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