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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimddvcbvw | Structured version Visualization version GIF version | ||
| Description: Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44218. The equivalent of this theorem without the bound variable change is rexlimddv 3161. Version of rexlimddvcbv 44220 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by GG, 2-Apr-2024.) | 
| Ref | Expression | 
|---|---|
| rexlimddvcbvw.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) | 
| rexlimddvcbvw.2 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | 
| rexlimddvcbvw.3 | ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rexlimddvcbvw | ⊢ (𝜑 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexlimddvcbvw.1 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) | |
| 2 | rexlimddvcbvw.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) | |
| 3 | 2 | cbvrexvw 3238 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜃 ↔ ∃𝑦 ∈ 𝐴 𝜒) | 
| 4 | rexlimddvcbvw.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) | |
| 5 | 4 | rexlimdvaa 3156 | . . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜒 → 𝜓)) | 
| 6 | 3, 5 | biimtrid 242 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜃 → 𝜓)) | 
| 7 | 1, 6 | mpd 15 | 1 ⊢ (𝜑 → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 | 
| 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-8 2110 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-clel 2816 df-rex 3071 | 
| This theorem is referenced by: mnuprdlem1 44291 mnuprdlem2 44292 | 
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