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Mirrors > Home > MPE Home > Th. List > cbvreuv | Structured version Visualization version GIF version |
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3386 for a version without ax-13 2372, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker cbvreuvw 3386 when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvralv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvreuv | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1917 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1917 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvralv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvreu 3381 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃!wreu 3066 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clel 2816 df-reu 3072 |
This theorem is referenced by: (None) |
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