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| Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker 2moswapv 2629 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| 2moswap | ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfe1 2150 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
| 2 | 1 | moexex 2638 | . . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) | 
| 3 | 2 | expcom 413 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) | 
| 4 | 19.8a 2181 | . . . . 5 ⊢ (𝜑 → ∃𝑦𝜑) | |
| 5 | 4 | pm4.71ri 560 | . . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑)) | 
| 6 | 5 | exbii 1848 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑)) | 
| 7 | 6 | mobii 2548 | . 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) | 
| 8 | 3, 7 | imbitrrdi 252 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∃*wmo 2538 | 
| 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 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 | 
| This theorem is referenced by: 2euswap 2645 | 
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