Theorem nfex 1648

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 1530	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	hbex 1647	. 2 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)
4	3	nfi 1473	1 ⊢ Ⅎ𝑥∃𝑦𝜑

Syntax hints:  Ⅎwnf 1471  ∃wex 1503

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  eeor  1706  cbvexv1  1763  cbvex2  1934  eean  1943  nfsbv  1959  nfeu1  2049  nfeuv  2056  nfel  2341  ceqsex2  2792  nfopab  4086  nfopab2  4088  cbvopab1  4091  cbvopab1s  4093  repizf2  4180  copsex2t  4263  copsex2g  4264  euotd  4272  onintrab2im  4535  mosubopt  4709  nfco  4810  dfdmf  4838  dfrnf  4886  nfdm  4889  fv3  5557  nfoprab2  5947  nfoprab3  5948  nfoprab  5949  cbvoprab1  5969  cbvoprab2  5970  cbvoprab3  5973  cnvoprab  6260  ac6sfi  6927  cc3  7298  nfsum1  11399  nfsum  11400  fsum2dlemstep  11477  nfcprod1  11597  nfcprod  11598  fprod2dlemstep  11665  lss1d  13716