New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > cbvreuv | GIF version |
Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
cbvralv.1 | ⊢ (x = y → (φ ↔ ψ)) |
Ref | Expression |
---|---|
cbvreuv | ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎyφ | |
2 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
3 | cbvralv.1 | . 2 ⊢ (x = y → (φ ↔ ψ)) | |
4 | 1, 2, 3 | cbvreu 2833 | 1 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∃!wreu 2616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-cleq 2346 df-clel 2349 df-reu 2621 |
This theorem is referenced by: reu8 3032 |
Copyright terms: Public domain | W3C validator |