Theorem nfex 1637

Description: If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)

Hypothesis

Ref	Expression
nfex.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfex	⊢ Ⅎ𝑥∃𝑦𝜑

Proof of Theorem nfex

Step	Hyp	Ref	Expression
1		nfex.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1519	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1636	. 2 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	3	nfi 1462	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:  Ⅎwnf 1460  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  eeor  1695  cbvexv1  1752  cbvex2  1922  eean  1931  nfsbv  1947  nfeu1  2037  nfeuv  2044  nfel  2328  ceqsex2  2777  nfopab  4070  nfopab2  4072  cbvopab1  4075  cbvopab1s  4077  repizf2  4161  copsex2t  4244  copsex2g  4245  euotd  4253  onintrab2im  4516  mosubopt  4690  nfco  4791  dfdmf  4819  dfrnf  4867  nfdm  4870  fv3  5537  nfoprab2  5922  nfoprab3  5923  nfoprab  5924  cbvoprab1  5944  cbvoprab2  5945  cbvoprab3  5948  cnvoprab  6232  ac6sfi  6895  cc3  7264  nfsum1  11357  nfsum  11358  fsum2dlemstep  11435  nfcprod1  11555  nfcprod  11556  fprod2dlemstep  11623