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| Mirrors > Home > MPE Home > Th. List > Mathboxes > exlimddvfi | Structured version Visualization version GIF version | ||
| Description: A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.) |
| Ref | Expression |
|---|---|
| exlimddvfi.1 | ⊢ (𝜑 → ∃𝑥𝜃) |
| exlimddvfi.2 | ⊢ Ⅎ𝑦𝜃 |
| exlimddvfi.3 | ⊢ Ⅎ𝑦𝜓 |
| exlimddvfi.4 | ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂) |
| exlimddvfi.5 | ⊢ ((𝜂 ∧ 𝜓) → 𝜒) |
| exlimddvfi.6 | ⊢ Ⅎ𝑦𝜒 |
| Ref | Expression |
|---|---|
| exlimddvfi | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | exlimddvfi.1 | . . 3 ⊢ (𝜑 → ∃𝑥𝜃) | |
| 2 | exlimddvfi.2 | . . . 4 ⊢ Ⅎ𝑦𝜃 | |
| 3 | 2 | sb8e 2523 | . . 3 ⊢ (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃) |
| 4 | 1, 3 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃) |
| 5 | exlimddvfi.3 | . 2 ⊢ Ⅎ𝑦𝜓 | |
| 6 | sbsbc 3792 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜃 ↔ [𝑦 / 𝑥]𝜃) | |
| 7 | exlimddvfi.4 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂) | |
| 8 | 6, 7 | bitri 275 | . . 3 ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂) |
| 9 | exlimddvfi.5 | . . 3 ⊢ ((𝜂 ∧ 𝜓) → 𝜒) | |
| 10 | 8, 9 | sylanb 581 | . 2 ⊢ (([𝑦 / 𝑥]𝜃 ∧ 𝜓) → 𝜒) |
| 11 | exlimddvfi.6 | . 2 ⊢ Ⅎ𝑦𝜒 | |
| 12 | 4, 5, 10, 11 | exlimddvf 38128 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1779 Ⅎwnf 1783 [wsb 2064 [wsbc 3788 |
| 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 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 |
| This theorem is referenced by: (None) |
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