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Mirrors > Home > MPE Home > Th. List > Mathboxes > 19.12b | Structured version Visualization version GIF version |
Description: Version of 19.12vv 2359 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
19.12b.1 | ⊢ Ⅎ𝑦𝜑 |
19.12b.2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
19.12b | ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.12b.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | 19.21 2197 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)) |
3 | 2 | exbii 1839 | . 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓)) |
4 | 19.12b.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | nfal 2333 | . . 3 ⊢ Ⅎ𝑥∀𝑦𝜓 |
6 | 5 | 19.36 2222 | . 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓)) |
7 | 4 | 19.36 2222 | . . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
8 | 7 | albii 1811 | . . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓)) |
9 | 1 | nfal 2333 | . . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑 |
10 | 9 | 19.21 2197 | . . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓)) |
11 | 8, 10 | bitr2i 277 | . 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) |
12 | 3, 6, 11 | 3bitri 298 | 1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-nf 1776 |
This theorem is referenced by: (None) |
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