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Mirrors > Home > ILE Home > Th. List > nfre1 | GIF version |
Description: 𝑥 is not free in ∃𝑥 ∈ 𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfre1 | ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2474 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | nfe1 1507 | . 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) | |
3 | 1, 2 | nfxfr 1485 | 1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 Ⅎwnf 1471 ∃wex 1503 ∈ wcel 2160 ∃wrex 2469 |
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-5 1458 ax-gen 1460 ax-ie1 1504 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-rex 2474 |
This theorem is referenced by: r19.29an 2632 nfiu1 3931 fun11iun 5501 eusvobj2 5881 fodjuomnilemdc 7171 ismkvnex 7182 prarloclem3step 7524 prmuloc2 7595 ltexprlemm 7628 caucvgprprlemaddq 7736 caucvgsrlemgt1 7823 axpre-suploclemres 7929 supinfneg 9624 infsupneg 9625 lbzbi 9645 divalglemeunn 11957 divalglemeuneg 11959 bezoutlemmain 12030 bezout 12043 lss1d 13696 pw1nct 15206 isomninnlem 15232 trirec0 15246 ismkvnnlem 15254 |
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