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  3233  hbra1  3273  nfralw  3283  bnj1316  34783  bnj1379  34793  bnj1468  34809  bnj958  34903  bnj981  34913  bnj1014  34924  bnj1128  34953  bnj1204  34975  bnj1279  34981  bnj1398  34997  bnj1408  34999  bnj1444  35006  bnj1445  35007  bnj1446  35008  bnj1447  35009  bnj1448  35010  bnj1449  35011  bnj1463  35018  bnj1312  35021  bnj1518  35027  bnj1519  35028  bnj1520  35029  bnj1525  35032  bj-cbv2v  36759  bj-equs45fv  36772  mpobi123f  38129