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  2403  equsalh  2419  equsexh  2420  nfae  2432  axc16i  2435  dvelimh  2449  nfs1  2487  hbsb  2523  sb7h  2525  nfsab  2720  nfsabg  2721  cleqh  2858  nfcii  2881  nfralw  3287  bnj596  34743  bnj1146  34788  bnj1379  34827  bnj1464  34841  bnj1468  34843  bnj605  34904  bnj607  34913  bnj916  34930  bnj964  34940  bnj981  34947  bnj983  34948  bnj1014  34958  bnj1123  34983  bnj1373  35027  bnj1417  35038  bnj1445  35041  bnj1463  35052  bnj1497  35057  bj-cbv3hv2  36790  bj-equsalhv  36801  bj-nfs1v  36808  bj-nfsab1  36811  bj-gabima  36935  wl-nfalv  37520  nfequid-o  38910  nfa1-o  38915  2sb5ndVD  44906  2sb5ndALT  44928