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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 805 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | nfdc.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | 2 | nfn 1621 | . . 3 ⊢ Ⅎ𝑥 ¬ 𝜑 |
4 | 2, 3 | nfor 1538 | . 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑) |
5 | 1, 4 | nfxfr 1435 | 1 ⊢ Ⅎ𝑥DECID 𝜑 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∨ wo 682 DECID wdc 804 Ⅎwnf 1421 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-gen 1410 ax-ie2 1455 ax-4 1472 ax-ial 1499 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-tru 1319 df-fal 1322 df-nf 1422 |
This theorem is referenced by: 19.32dc 1642 finexdc 6764 ssfirab 6790 exfzdc 9985 nfsum1 11093 nfsum 11094 zsupcllemstep 11565 infssuzex 11569 ctiunctlemudc 11877 |
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