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. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbveuw 2686 when possible. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
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 2682 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑) |
3 | cbveu.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | cbveu.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbie 2540 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | eubii 2666 | . 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓) |
7 | 2, 6 | bitri 277 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1780 [wsb 2065 ∃!weu 2649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 |
This theorem is referenced by: cbvreu 3447 cbvreucsf 3926 |
Copyright terms: Public domain | W3C validator |