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  2778  nfopab  4072  nfopab2  4074  cbvopab1  4077  cbvopab1s  4079  repizf2  4163  copsex2t  4246  copsex2g  4247  euotd  4255  onintrab2im  4518  mosubopt  4692  nfco  4793  dfdmf  4821  dfrnf  4869  nfdm  4872  fv3  5539  nfoprab2  5925  nfoprab3  5926  nfoprab  5927  cbvoprab1  5947  cbvoprab2  5948  cbvoprab3  5951  cnvoprab  6235  ac6sfi  6898  cc3  7267  nfsum1  11364  nfsum  11365  fsum2dlemstep  11442  nfcprod1  11562  nfcprod  11563  fprod2dlemstep  11630