Theorem nfcnv 5223

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 5046	. 2 ⊢ ◡𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2		nfcv 2751	. . . 4 ⊢ Ⅎ𝑥𝑧
3		nfcnv.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		nfcv 2751	. . . 4 ⊢ Ⅎ𝑥𝑦
5	2, 3, 4	nfbr 4629	. . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦
6	5	nfopab 4650	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
7	1, 6	nfcxfr 2749	1 ⊢ Ⅎ𝑥◡𝐴

Syntax hints:  Ⅎwnfc 2738   class class class wbr 4583  {copab 4642  ^◡ccnv 5037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046

This theorem is referenced by:  nfrn  5289  nfpred  5602  nffun  5826  nff1  6012  nfsup  8240  nfinf  8271  gsumcom2  18197  ptbasfi  21194  mbfposr  23225  itg1climres  23287  funcnvmptOLD  28850  funcnvmpt  28851  nfwsuc  31008  aomclem8  36649  rfcnpre1  38201  rfcnpre2  38213