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Mirrors > Home > MPE Home > Th. List > exdistrf | Structured version Visualization version GIF version |
Description: Distribution of existential quantifiers, with a bound-variable hypothesis saying that 𝑦 is not free in 𝜑, but 𝑥 can be free in 𝜑 (and there is no distinct variable condition on 𝑥 and 𝑦). (Contributed by Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.) |
Ref | Expression |
---|---|
exdistrf.1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑) |
Ref | Expression |
---|---|
exdistrf | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2145 | . 2 ⊢ Ⅎ𝑥∃𝑥(𝜑 ∧ ∃𝑦𝜓) | |
2 | 19.8a 2170 | . . . . . 6 ⊢ (𝜓 → ∃𝑦𝜓) | |
3 | 2 | anim2i 616 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ ∃𝑦𝜓)) |
4 | 3 | eximi 1826 | . . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑦(𝜑 ∧ ∃𝑦𝜓)) |
5 | biidd 263 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝜑 ∧ ∃𝑦𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))) | |
6 | 5 | drex1 2455 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ ∃𝑦(𝜑 ∧ ∃𝑦𝜓))) |
7 | 4, 6 | syl5ibr 247 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))) |
8 | 19.40 1878 | . . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓)) | |
9 | exdistrf.1 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝜑) | |
10 | 9 | 19.9d 2193 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦𝜑 → 𝜑)) |
11 | 10 | anim1d 610 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑦𝜑 ∧ ∃𝑦𝜓) → (𝜑 ∧ ∃𝑦𝜓))) |
12 | 19.8a 2170 | . . . 4 ⊢ ((𝜑 ∧ ∃𝑦𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) | |
13 | 8, 11, 12 | syl56 36 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓))) |
14 | 7, 13 | pm2.61i 183 | . 2 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
15 | 1, 14 | exlimi 2207 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀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-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 |
This theorem is referenced by: oprabid 7177 |
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