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Mirrors > Home > MPE Home > Th. List > cbvmowOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cbvmow 2605 as of 23-May-2024. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvmowOLD.1 | ⊢ Ⅎ𝑦𝜑 |
cbvmowOLD.2 | ⊢ Ⅎ𝑥𝜓 |
cbvmowOLD.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvmowOLD | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvmowOLD.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sb8ev 2355 | . . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) |
3 | 1 | sb8euv 2601 | . . . 4 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
4 | 2, 3 | imbi12i 351 | . . 3 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)) |
5 | moeu 2585 | . . 3 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
6 | moeu 2585 | . . 3 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑)) | |
7 | 4, 5, 6 | 3bitr4i 303 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) |
8 | cbvmowOLD.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
9 | cbvmowOLD.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | sbiev 2313 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
11 | 10 | mobii 2550 | . 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓) |
12 | 7, 11 | bitri 274 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wex 1786 Ⅎwnf 1790 [wsb 2071 ∃*wmo 2540 ∃!weu 2570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 |
This theorem is referenced by: (None) |
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