Theorem nfcnv 5210

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 5035	. 2 ⊢ ◡𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2		nfcv 2750	. . . 4 ⊢ Ⅎ𝑥𝑧
3		nfcnv.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2750	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4623	. . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦
6	5	nfopab 4644	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
7	1, 6	nfcxfr 2748	1 ⊢ Ⅎ𝑥◡𝐴

Syntax hints:  Ⅎwnfc 2737   class class class wbr 4577  {copab 4636  ^◡ccnv 5026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cnv 5035

This theorem is referenced by:  nfrn  5275  nfpred  5587  nffun  5811  nff1  5996  nfsup  8217  nfinf  8248  gsumcom2  18145  ptbasfi  21141  mbfposr  23169  itg1climres  23231  funcnvmptOLD  28643  funcnvmpt  28644  nfwsuc  30801  aomclem8  36432  rfcnpre1  37984  rfcnpre2  37996