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Theorem nfcnv 4762
 Description: Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
nfcnv.1 𝑥𝐴
Assertion
Ref Expression
nfcnv 𝑥𝐴

Proof of Theorem nfcnv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4591 . 2 𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2 nfcv 2299 . . . 4 𝑥𝑧
3 nfcnv.1 . . . 4 𝑥𝐴
4 nfcv 2299 . . . 4 𝑥𝑦
52, 3, 4nfbr 4010 . . 3 𝑥 𝑧𝐴𝑦
65nfopab 4032 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
71, 6nfcxfr 2296 1 𝑥𝐴
 Colors of variables: wff set class Syntax hints:  Ⅎwnfc 2286   class class class wbr 3965  {copab 4024  ◡ccnv 4582 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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4591 This theorem is referenced by:  nfrn  4828  nffun  5190  nff1  5370  nfinf  6953
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