Theorem nfex 1685

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 1567	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1684	. 2 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	3	nfi 1510	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:  Ⅎwnf 1508  ∃wex 1540

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558

This theorem depends on definitions:  df-bi 117  df-nf 1509

This theorem is referenced by:  eeor  1743  cbvexv1  1800  cbvex2  1971  eean  1984  nfsbv  2000  nfeu1  2090  nfeuv  2097  nfel  2383  ceqsex2  2844  nfopab  4157  nfopab2  4159  cbvopab1  4162  cbvopab1s  4164  repizf2  4252  copsex2t  4337  copsex2g  4338  euotd  4347  onintrab2im  4616  mosubopt  4791  nfco  4895  dfdmf  4924  dfrnf  4973  nfdm  4976  fv3  5662  nfoprab2  6071  nfoprab3  6072  nfoprab  6073  cbvoprab1  6093  cbvoprab2  6094  cbvoprab3  6097  cnvoprab  6399  ac6sfi  7087  cc3  7487  nfsum1  11921  nfsum  11922  fsum2dlemstep  12000  nfcprod1  12120  nfcprod  12121  fprod2dlemstep  12188  lss1d  14403