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Mathbox for Giovanni Mascellani |
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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 2518 | . . 3 ⊢ (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃) |
4 | 1, 3 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃) |
5 | exlimddvfi.3 | . 2 ⊢ Ⅎ𝑦𝜓 | |
6 | sbsbc 3781 | . . . 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 582 | . 2 ⊢ (([𝑦 / 𝑥]𝜃 ∧ 𝜓) → 𝜒) |
11 | exlimddvfi.6 | . 2 ⊢ Ⅎ𝑦𝜒 | |
12 | 4, 5, 10, 11 | exlimddvf 36978 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∃wex 1782 Ⅎwnf 1786 [wsb 2068 [wsbc 3777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2372 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3778 |
This theorem is referenced by: (None) |
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