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Mirrors > Home > MPE Home > Th. List > 19.12vv | Structured version Visualization version GIF version |
Description: Special case of 19.12 2325 where its converse holds. See 19.12vvv 2074 for a version with a dv 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 2019 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)) | |
2 | 1 | exbii 1923 | . 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓)) |
3 | nfv 1994 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfal 2316 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 |
5 | 4 | 19.36 2253 | . 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓)) |
6 | 19.36v 2071 | . . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) | |
7 | 6 | albii 1894 | . . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓)) |
8 | nfv 1994 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
9 | 8 | nfal 2316 | . . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑 |
10 | 9 | 19.21 2230 | . . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓)) |
11 | 7, 10 | bitr2i 265 | . 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) |
12 | 2, 5, 11 | 3bitri 286 | 1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1628 ∃wex 1851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-10 2173 ax-11 2189 ax-12 2202 |
This theorem depends on definitions: df-bi 197 df-ex 1852 df-nf 1857 |
This theorem is referenced by: (None) |
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