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Mirrors > Home > MPE Home > Th. List > cbveuwOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cbveuw 2626 as of 23-May-2024. (Contributed by NM, 25-Nov-1994.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbveuwOLD.1 | ⊢ Ⅎ𝑦𝜑 |
cbveuwOLD.2 | ⊢ Ⅎ𝑥𝜓 |
cbveuwOLD.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbveuwOLD | ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbveuwOLD.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sb8euv 2620 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
3 | cbveuwOLD.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | cbveuwOLD.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbiev 2323 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | eubii 2605 | . 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓) |
7 | 2, 6 | bitri 278 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1786 [wsb 2070 ∃!weu 2588 |
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 1912 ax-6 1971 ax-7 2016 ax-10 2143 ax-11 2159 ax-12 2176 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 |
This theorem is referenced by: (None) |
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