Theorem nfsuc 6432

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 6363	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		nfsuc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	nfsn 4668	. . 3 ⊢ Ⅎ𝑥{𝐴}
4	2, 3	nfun 4117	. 2 ⊢ Ⅎ𝑥(𝐴 ∪ {𝐴})
5	1, 4	nfcxfr 2920	1 ⊢ Ⅎ𝑥 suc 𝐴

This proof depends on syntax axioms:  Ⅎwnfc 2907   ∪ cun 3897  {csn 4584  suc csuc 6359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-un 3904  df-sn 4585  df-pr 4587  df-suc 6363

This theorem is used by:  ttrcltr  9695  rankidb  9782  dfon2lem3  36362