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Mirrors > Home > ILE Home > Th. List > nfcxfrd | GIF version |
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfceqi.1 | ⊢ 𝐴 = 𝐵 |
nfcxfrd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfcxfrd | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcxfrd.2 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
2 | nfceqi.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | nfceqi 2249 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
4 | 1, 3 | sylibr 133 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 Ⅎwnfc 2240 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-4 1468 ax-17 1487 ax-ial 1495 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-cleq 2106 df-clel 2109 df-nfc 2242 |
This theorem is referenced by: nfcsb1d 2997 nfcsbd 3000 nfifd 3463 nfunid 3707 nfiotadxy 5047 nfriotadxy 5690 nfovd 5752 nfnegd 7875 |
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