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Theorem nfcnv 5333
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 5151 . 2 𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2 nfcv 2793 . . . 4 𝑥𝑧
3 nfcnv.1 . . . 4 𝑥𝐴
4 nfcv 2793 . . . 4 𝑥𝑦
52, 3, 4nfbr 4732 . . 3 𝑥 𝑧𝐴𝑦
65nfopab 4751 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
71, 6nfcxfr 2791 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wnfc 2780   class class class wbr 4685  {copab 4745  ccnv 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cnv 5151
This theorem is referenced by:  nfrn  5400  nfpred  5723  nffun  5949  nff1  6137  nfsup  8398  nfinf  8429  gsumcom2  18420  ptbasfi  21432  mbfposr  23464  itg1climres  23526  funcnvmptOLD  29595  funcnvmpt  29596  nfwsuc  31888  aomclem8  37948  rfcnpre1  39492  rfcnpre2  39504  smfpimcc  41335
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