Theorem nfpred 5685

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 5680	. 2 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2		nfpred.2	. . 3 ⊢ Ⅎ𝑥𝐴
3		nfpred.1	. . . . 5 ⊢ Ⅎ𝑥𝑅
4	3	nfcnv 5301	. . . 4 ⊢ Ⅎ𝑥◡𝑅
5		nfpred.3	. . . . 5 ⊢ Ⅎ𝑥𝑋
6	5	nfsn 4242	. . . 4 ⊢ Ⅎ𝑥{𝑋}
7	4, 6	nfima 5474	. . 3 ⊢ Ⅎ𝑥(◡𝑅 “ {𝑋})
8	2, 7	nfin 3820	. 2 ⊢ Ⅎ𝑥(𝐴 ∩ (◡𝑅 “ {𝑋}))
9	1, 8	nfcxfr 2762	1 ⊢ Ⅎ𝑥Pred(𝑅, 𝐴, 𝑋)

Syntax hints:  Ⅎwnfc 2751   ∩ cin 3573  {csn 4177  ^◡ccnv 5113   “ cima 5117  Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  nfwrecs  7409  nfwsuc  31764  nfwlim  31768