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| Description: Equality theorem for the nonfreeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1784 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 1851 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)) | |
| 2 | nfbii.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
| 3 | 1, 2 | mpg 1797 | 1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 Ⅎwnf 1783 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 | 
| This theorem is referenced by: nfxfr 1853 nfxfrd 1854 dvelimhw 2347 nfeqf1 2384 nfceqi 2902 nfra2wOLD 3300 dfnfc2 4929 nfan1c 35087 bj-dvelimdv1 36853 bj-nfcf 36924 iunconnlem2 44955 | 
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