Theorem nfpred 6300

Description: Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.)

Hypotheses

Ref	Expression
nfpred.1	⊢ Ⅎ𝑥𝑅
nfpred.2	⊢ Ⅎ𝑥𝐴
nfpred.3	⊢ Ⅎ𝑥𝑋

Assertion

Ref	Expression
nfpred	⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋)

Proof of Theorem nfpred

Step	Hyp	Ref	Expression
1		df-pred 6295	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		nfpred.2	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfpred.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
4	3	nfcnv 5863	. . . 4 ⊢ Ⅎ𝑥◡𝑅
5		nfpred.3	. . . . 5 ⊢ Ⅎ𝑥𝑋
6	5	nfsn 4688	. . . 4 ⊢ Ⅎ𝑥{𝑋}
7	4, 6	nfima 6060	. . 3 ⊢ Ⅎ𝑥(◡𝑅 “ {𝑋})
8	2, 7	nfin 4204	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (◡𝑅 “ {𝑋}))
9	1, 8	nfcxfr 2897	1 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋)

Syntax hints:  Ⅎwnfc 2884   ∩ cin 3930  {csn 4606  ^◡ccnv 5658   “ cima 5662  Predcpred 6294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295

This theorem is referenced by:  nffrecs  8287  nfwrecsOLD  8321  nfwsuc  35841  nfwlim  35845