![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cbveu | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
cbveu.1 | ⊢ Ⅎ𝑦𝜑 |
cbveu.2 | ⊢ Ⅎ𝑥𝜓 |
cbveu.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbveu | ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbveu.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sb8eu 2688 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
3 | cbveu.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | cbveu.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbie 2539 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | eubii 2658 | . 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓) |
7 | 2, 6 | bitri 267 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 Ⅎwnf 1884 [wsb 2069 ∃!weu 2639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 |
This theorem is referenced by: cbvmoOLD 2693 cbvreu 3381 cbvreucsf 3791 tz6.12f 6457 f1ompt 6630 climeu 14663 initoeu2 17018 |
Copyright terms: Public domain | W3C validator |