Theorem nf5i 2147

Description: Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nf5i.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
nf5i	⊢ Ⅎ𝑥𝜑

Proof of Theorem nf5i

Step	Hyp	Ref	Expression
1		nf5-1 2146	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑)
2		nf5i.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	mpg 1797	1 ⊢ Ⅎ𝑥𝜑

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-10 2142

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

This theorem is referenced by:  nfnaew  2150  nfe1  2151  sbh  2273  nf5di  2285  19.9h  2286  19.21h  2287  19.23h  2288  exlimih  2289  exlimdh  2290  equsalhw  2291  equsexhv  2292  hban  2300  hb3an  2301  nfal  2322  hbex  2324  nfsbv  2329  cbv3hv  2338  dvelimhw  2343  cbv3h  2402  equsalh  2418  equsexh  2419  nfae  2431  axc16i  2434  dvelimh  2448  nfs1  2486  hbsb  2522  sb7h  2524  nfsab  2719  nfsabg  2720  cleqh  2857  nfcii  2880  nfralw  3283  bnj596  34729  bnj1146  34774  bnj1379  34813  bnj1464  34827  bnj1468  34829  bnj605  34890  bnj607  34899  bnj916  34916  bnj964  34926  bnj981  34933  bnj983  34934  bnj1014  34944  bnj1123  34969  bnj1373  35013  bnj1417  35024  bnj1445  35027  bnj1463  35038  bnj1497  35043  bj-cbv3hv2  36776  bj-equsalhv  36787  bj-nfs1v  36794  bj-nfsab1  36797  bj-gabima  36921  wl-nfalv  37506  nfequid-o  38896  nfa1-o  38901  2sb5ndVD  44892  2sb5ndALT  44914