Theorem nfsuc 6380

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 6312	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		nfsuc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	nfsn 4660	. . 3 ⊢ Ⅎ𝑥{𝐴}
4	2, 3	nfun 4120	. 2 ⊢ Ⅎ𝑥(𝐴 ∪ {𝐴})
5	1, 4	nfcxfr 2892	1 ⊢ Ⅎ𝑥 suc 𝐴

Syntax hints:  Ⅎwnfc 2879   ∪ cun 3900  {csn 4576  suc csuc 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-v 3438  df-un 3907  df-sn 4577  df-pr 4579  df-suc 6312

This theorem is referenced by:  ttrcltr  9606  rankidb  9693  dfon2lem3  35825