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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm11.58 | Structured version Visualization version GIF version |
Description: Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
pm11.58 | ⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.8a 2180 | . . . . 5 ⊢ (𝜑 → ∃𝑥𝜑) | |
2 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | sb8e 2560 | . . . . 5 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
4 | 1, 3 | sylib 220 | . . . 4 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜑) |
5 | 4 | pm4.71i 562 | . . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑)) |
6 | 19.42v 1954 | . . 3 ⊢ (∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ ∃𝑦[𝑦 / 𝑥]𝜑)) | |
7 | 5, 6 | bitr4i 280 | . 2 ⊢ (𝜑 ↔ ∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
8 | 7 | exbii 1848 | 1 ⊢ (∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝜑 ∧ [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∃wex 1780 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: (None) |
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