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Theorem nfre1 2509
Description: 𝑥 is not free in 𝑥𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
Assertion
Ref Expression
nfre1 𝑥𝑥𝐴 𝜑

Proof of Theorem nfre1
StepHypRef Expression
1 df-rex 2450 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 nfe1 1484 . 2 𝑥𝑥(𝑥𝐴𝜑)
31, 2nfxfr 1462 1 𝑥𝑥𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wa 103  wnf 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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