Theorem nfex 1625

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 1507	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1624	. 2 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	3	nfi 1450	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:  Ⅎwnf 1448  ∃wex 1480

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  eeor  1683  cbvexv1  1740  cbvex2  1910  eean  1919  nfsbv  1935  nfeu1  2025  nfeuv  2032  nfel  2317  ceqsex2  2766  nfopab  4050  nfopab2  4052  cbvopab1  4055  cbvopab1s  4057  repizf2  4141  copsex2t  4223  copsex2g  4224  euotd  4232  onintrab2im  4495  mosubopt  4669  nfco  4769  dfdmf  4797  dfrnf  4845  nfdm  4848  fv3  5509  nfoprab2  5892  nfoprab3  5893  nfoprab  5894  cbvoprab1  5914  cbvoprab2  5915  cbvoprab3  5918  cnvoprab  6202  ac6sfi  6864  cc3  7209  nfsum1  11297  nfsum  11298  fsum2dlemstep  11375  nfcprod1  11495  nfcprod  11496  fprod2dlemstep  11563