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Mirrors > Home > MPE Home > Th. List > nfcxfrd | Structured version Visualization version 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 2967 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
4 | 1, 3 | sylibr 226 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 Ⅎwnfc 2957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1662 df-ex 1881 df-nf 1885 df-cleq 2819 df-clel 2822 df-nfc 2959 |
This theorem is referenced by: nfcsb1d 3772 nfcsbd 3775 nfifd 4335 nfunid 4666 nfiotad 6090 nfriotad 6875 nfovd 6935 nfnegd 10597 nfxnegd 40464 nfintd 43316 nfiund 43317 |
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