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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 2450 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | nfe1 1484 | . 2 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) | |
3 | 1, 2 | nfxfr 1462 | 1 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 Ⅎwnf 1448 ∃wex 1480 ∈ wcel 2136 ∃wrex 2445 |
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-5 1435 ax-gen 1437 ax-ie1 1481 |
This theorem depends on definitions: df-bi 116 df-nf 1449 df-rex 2450 |
This theorem is referenced by: r19.29an 2608 nfiu1 3896 fun11iun 5453 eusvobj2 5828 fodjuomnilemdc 7108 ismkvnex 7119 prarloclem3step 7437 prmuloc2 7508 ltexprlemm 7541 caucvgprprlemaddq 7649 caucvgsrlemgt1 7736 axpre-suploclemres 7842 supinfneg 9533 infsupneg 9534 lbzbi 9554 divalglemeunn 11858 divalglemeuneg 11860 bezoutlemmain 11931 bezout 11944 pw1nct 13883 isomninnlem 13909 trirec0 13923 ismkvnnlem 13931 |
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