Theorem nfsuc 6439

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 6370	. 2 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		nfsuc.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	2	nfsn 4675	. . 3 ⊢ Ⅎ𝑥{𝐴}
4	2, 3	nfun 4124	. 2 ⊢ Ⅎ𝑥(𝐴 ∪ {𝐴})
5	1, 4	nfcxfr 2925	1 ⊢ Ⅎ𝑥 suc 𝐴

This proof depends on syntax axioms:  Ⅎwnfc 2912   ∪ cun 3904  {csn 4591  suc csuc 6366

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-v 3459  df-un 3911  df-sn 4592  df-pr 4594  df-suc 6370

This theorem is used by:  ttrcltr  9688  rankidb  9775  dfon2lem3  36288