Theorem nfex 1528

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 1412	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1527	. 2 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	3	nfi 1351	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:  Ⅎwnf 1349  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  eeor  1585  cbvex2  1797  eean  1806  nfeu1  1911  nfeuv  1918  nfel  2186  ceqsex2  2594  nfopab  3825  nfopab2  3827  cbvopab1  3830  cbvopab1s  3832  repizf2  3915  copsex2t  3982  copsex2g  3983  euotd  3991  onintrab2im  4244  mosubopt  4405  nfco  4501  dfdmf  4528  dfrnf  4575  nfdm  4578  fv3  5197  nfoprab2  5555  nfoprab3  5556  nfoprab  5557  cbvoprab1  5576  cbvoprab2  5577  cbvoprab3  5580  ac6sfi  6352  nfsum1  9875  nfsum  9876