Theorem nfex 1616

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 1499	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1615	. 2 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	3	nfi 1438	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:  Ⅎwnf 1436  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  eeor  1673  cbvex2  1894  eean  1903  nfeu1  2010  nfeuv  2017  nfel  2290  ceqsex2  2726  nfopab  3996  nfopab2  3998  cbvopab1  4001  cbvopab1s  4003  repizf2  4086  copsex2t  4167  copsex2g  4168  euotd  4176  onintrab2im  4434  mosubopt  4604  nfco  4704  dfdmf  4732  dfrnf  4780  nfdm  4783  fv3  5444  nfoprab2  5821  nfoprab3  5822  nfoprab  5823  cbvoprab1  5843  cbvoprab2  5844  cbvoprab3  5847  cnvoprab  6131  ac6sfi  6792  nfsum1  11125  nfsum  11126  fsum2dlemstep  11203  nfcprod1  11323  nfcprod  11324