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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 821 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | nfdc.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | nfn 1638 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
4 | 2, 3 | nfor 1554 | . 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑) |
5 | 1, 4 | nfxfr 1454 | 1 ⊢ Ⅎ𝑥DECID 𝜑 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 698 DECID wdc 820 Ⅎwnf 1440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-gen 1429 ax-ie2 1474 ax-4 1490 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1338 df-fal 1341 df-nf 1441 |
This theorem is referenced by: 19.32dc 1659 finexdc 6847 ssfirab 6878 dcfi 6925 exfzdc 10139 nfsum1 11253 nfsum 11254 nfcprod1 11451 nfcprod 11452 zsupcllemstep 11832 infssuzex 11836 ctiunctlemudc 12177 iswomninnlem 13631 |
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