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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfcxfrdf | Structured version Visualization version GIF version | ||
| Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by NM, 19-Nov-2020.) | 
| Ref | Expression | 
|---|---|
| nfcxfrdf.0 | ⊢ Ⅎ𝑥𝜑 | 
| nfcxfrdf.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| nfcxfrdf.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) | 
| Ref | Expression | 
|---|---|
| nfcxfrdf | ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfcxfrdf.2 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 2 | nfcxfrdf.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 3 | nfcxfrdf.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 4 | 2, 3 | nfceqdf 2901 | . 2 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) | 
| 5 | 1, 4 | mpbird 257 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 Ⅎwnf 1783 Ⅎwnfc 2890 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-cleq 2729 df-nfc 2892 | 
| This theorem is referenced by: (None) | 
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