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Mirrors > Home > ILE Home > Th. List > nfdc | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in DECID 𝜑. (Contributed by Jim Kingdon, 11-Mar-2018.) |
Ref | Expression |
---|---|
nfdc.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfdc | ⊢ Ⅎ𝑥DECID 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 836 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | nfdc.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | nfn 1668 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
4 | 2, 3 | nfor 1584 | . 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑) |
5 | 1, 4 | nfxfr 1484 | 1 ⊢ Ⅎ𝑥DECID 𝜑 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 709 DECID wdc 835 Ⅎwnf 1470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-gen 1459 ax-ie2 1504 ax-4 1520 ax-ial 1544 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1366 df-fal 1369 df-nf 1471 |
This theorem is referenced by: 19.32dc 1689 finexdc 6916 ssfirab 6947 dcfi 6994 exfzdc 10254 nfsum1 11378 nfsum 11379 nfcprod1 11576 nfcprod 11577 zsupcllemstep 11960 infssuzex 11964 nnwosdc 12054 ctiunctlemudc 12452 iswomninnlem 15094 |
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