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	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcnv	⊢ Ⅎ𝑥◡𝐴

Proof of Theorem nfcnv

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cnv 5151	. 2 ⊢ ◡𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2		nfcv 2793	. . . 4 ⊢ Ⅎ𝑥𝑧
3		nfcnv.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2793	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4732	. . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦
6	5	nfopab 4751	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
7	1, 6	nfcxfr 2791	1 ⊢ Ⅎ𝑥◡𝐴

Syntax hints:  Ⅎwnfc 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