![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfbii | Structured version Visualization version GIF version |
Description: Equality theorem for the non-freeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 12-Sep-2021.) |
Ref | Expression |
---|---|
nfbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
nfbii | ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfbiit 1852 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)) | |
2 | nfbii.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
3 | 1, 2 | mpg 1799 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 210 df-ex 1782 df-nf 1786 |
This theorem is referenced by: nfxfr 1854 nfxfrd 1855 dvelimhw 2355 nfeqf1 2386 nfceqi 2952 dfnfc2 4822 bj-dvelimdv1 34291 bj-nfcf 34366 iunconnlem2 41641 |
Copyright terms: Public domain | W3C validator |