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| Mirrors > Home > MPE Home > Th. List > Mathboxes > e2ebind | Structured version Visualization version GIF version | ||
| Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 44583 is derived from e2ebindVD 44932. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| e2ebind | ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | biidd 262 | . . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝜑 ↔ 𝜑)) | |
| 2 | 1 | drex1 2446 | . . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑)) |
| 3 | 2 | drex2 2447 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)) |
| 4 | excom 2162 | . . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) | |
| 5 | 3, 4 | bitrdi 287 | . . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑)) |
| 6 | nfe1 2150 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
| 7 | 6 | 19.9 2205 | . . 3 ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑) |
| 8 | 5, 7 | bitr3di 286 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
| 9 | 8 | aecoms 2433 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 ∃wex 1779 |
| 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 |
| This theorem is referenced by: (None) |
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