Theorem nfsuc 6406

Description: Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)

Hypothesis

Ref	Expression
nfsuc.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfsuc	⊢ Ⅎ𝑥 suc 𝐴

Proof of Theorem nfsuc

Step	Hyp	Ref	Expression
1		df-suc 6338	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		nfsuc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	nfsn 4671	. . 3 ⊢ Ⅎ𝑥{𝐴}
4	2, 3	nfun 4133	. 2 ⊢ Ⅎ𝑥(𝐴 ∪ {𝐴})
5	1, 4	nfcxfr 2889	1 ⊢ Ⅎ𝑥 suc 𝐴

Syntax hints:  Ⅎwnfc 2876   ∪ cun 3912  {csn 4589  suc csuc 6334

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 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-v 3449  df-un 3919  df-sn 4590  df-pr 4592  df-suc 6338

This theorem is referenced by:  ttrcltr  9669  rankidb  9753  dfon2lem3  35773