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| Description: Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2141. (Revised by GG, 21-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| eeor.1 | ⊢ Ⅎ𝑦𝜑 | 
| eeor.2 | ⊢ Ⅎ𝑥𝜓 | 
| Ref | Expression | 
|---|---|
| eeor | ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 19.43 1882 | . . 3 ⊢ (∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑦𝜑 ∨ ∃𝑦𝜓)) | |
| 2 | 1 | exbii 1848 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ ∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓)) | 
| 3 | 19.43 1882 | . . 3 ⊢ (∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥∃𝑦𝜑 ∨ ∃𝑥∃𝑦𝜓)) | |
| 4 | eeor.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
| 5 | 4 | 19.9 2205 | . . . . 5 ⊢ (∃𝑦𝜑 ↔ 𝜑) | 
| 6 | 5 | exbii 1848 | . . . 4 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥𝜑) | 
| 7 | excom 2162 | . . . . 5 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦∃𝑥𝜓) | |
| 8 | eeor.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝜓 | |
| 9 | 8 | 19.9 2205 | . . . . . 6 ⊢ (∃𝑥𝜓 ↔ 𝜓) | 
| 10 | 9 | exbii 1848 | . . . . 5 ⊢ (∃𝑦∃𝑥𝜓 ↔ ∃𝑦𝜓) | 
| 11 | 7, 10 | bitri 275 | . . . 4 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓) | 
| 12 | 6, 11 | orbi12i 915 | . . 3 ⊢ ((∃𝑥∃𝑦𝜑 ∨ ∃𝑥∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) | 
| 13 | 3, 12 | bitri 275 | . 2 ⊢ (∃𝑥(∃𝑦𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) | 
| 14 | 2, 13 | bitri 275 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 848 ∃wex 1779 Ⅎwnf 1783 | 
| 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-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-or 849 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: (None) | 
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