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Mirrors > Home > ILE Home > Th. List > nfnf | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in Ⅎ𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
Ref | Expression |
---|---|
nfal.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfnf | ⊢ Ⅎ𝑥Ⅎ𝑦𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nf 1395 | . 2 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑦(𝜑 → ∀𝑦𝜑)) | |
2 | nfal.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | nfal 1513 | . . . 4 ⊢ Ⅎ𝑥∀𝑦𝜑 |
4 | 2, 3 | nfim 1509 | . . 3 ⊢ Ⅎ𝑥(𝜑 → ∀𝑦𝜑) |
5 | 4 | nfal 1513 | . 2 ⊢ Ⅎ𝑥∀𝑦(𝜑 → ∀𝑦𝜑) |
6 | 1, 5 | nfxfr 1408 | 1 ⊢ Ⅎ𝑥Ⅎ𝑦𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1287 Ⅎwnf 1394 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-7 1382 ax-gen 1383 ax-4 1445 ax-ial 1472 ax-i5r 1473 |
This theorem depends on definitions: df-bi 115 df-nf 1395 |
This theorem is referenced by: nfnfc 2235 |
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