Theorem nf5ri 2196

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 15-Mar-2023.)

Hypothesis

Ref	Expression
nf5ri.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nf5ri	⊢ (𝜑 → ∀𝑥𝜑)

Proof of Theorem nf5ri

Step	Hyp	Ref	Expression
1		nf5ri.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1789	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
3	2	19.23bi 2192	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  19.3  2203  alimd  2213  alrimi  2214  eximd  2217  nexd  2222  albid  2223  exbid  2224  hbs1  2274  hba1  2293  hban  2300  hb3an  2301  nfal  2322  hbex  2324  nfsbv  2329  cbv3v  2333  cbv3  2395  equs45f  2457  nfs1  2486  sb6f  2495  hbsb  2522  hbab1  2716  nfsab  2719  nfsabg  2720  nfcrii  2886  ralrimi  3235  hbra1  3275  nfralw  3285  bnj1316  34810  bnj1379  34820  bnj1468  34836  bnj958  34930  bnj981  34940  bnj1014  34951  bnj1128  34980  bnj1204  35002  bnj1279  35008  bnj1398  35024  bnj1408  35026  bnj1444  35033  bnj1445  35034  bnj1446  35035  bnj1447  35036  bnj1448  35037  bnj1449  35038  bnj1463  35045  bnj1312  35048  bnj1518  35054  bnj1519  35055  bnj1520  35056  bnj1525  35059  bj-cbv2v  36786  bj-equs45fv  36799  mpobi123f  38156