Theorem nfpred 6270

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 6265	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		nfpred.2	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfpred.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
4	3	nfcnv 5833	. . . 4 ⊢ Ⅎ𝑥◡𝑅
5		nfpred.3	. . . . 5 ⊢ Ⅎ𝑥𝑋
6	5	nfsn 4651	. . . 4 ⊢ Ⅎ𝑥{𝑋}
7	4, 6	nfima 6033	. . 3 ⊢ Ⅎ𝑥(◡𝑅 “ {𝑋})
8	2, 7	nfin 4164	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (◡𝑅 “ {𝑋}))
9	1, 8	nfcxfr 2896	1 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋)

Syntax hints:  Ⅎwnfc 2883   ∩ cin 3888  {csn 4567  ^◡ccnv 5630   “ cima 5634  Predcpred 6264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265

This theorem is referenced by:  nffrecs  8233  nfwsuc  35998  nfwlim  36002