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  1947  nfsbv  1963  nfeu1  2053  nfeuv  2060  nfel  2345  ceqsex2  2800  nfopab  4097  nfopab2  4099  cbvopab1  4102  cbvopab1s  4104  repizf2  4191  copsex2t  4274  copsex2g  4275  euotd  4283  onintrab2im  4550  mosubopt  4724  nfco  4827  dfdmf  4855  dfrnf  4903  nfdm  4906  fv3  5577  nfoprab2  5968  nfoprab3  5969  nfoprab  5970  cbvoprab1  5990  cbvoprab2  5991  cbvoprab3  5994  cnvoprab  6287  ac6sfi  6954  cc3  7328  nfsum1  11499  nfsum  11500  fsum2dlemstep  11577  nfcprod1  11697  nfcprod  11698  fprod2dlemstep  11765  lss1d  13879