Theorem nf5rd 2203

Description: Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nf5rd.1	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nf5rd	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Proof of Theorem nf5rd

Step	Hyp	Ref	Expression
1		nf5rd.1	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜓)
2		nf5r 2201	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1539  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2184

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-nf 1785

This theorem is referenced by:  spimedv  2204  alrimdd  2221  nf5di  2291  hbnt  2300  hbimd  2304  dvelimhw  2349  dveeq2  2382  dveeq1  2384  axc9  2386  spimed  2392  dvelimh  2454  abidnf  3660  eusvnfb  5338  axrepnd  10505  axacndlem4  10521  bj-cbv2v  36999  bj-elgab  37140  wl-nfeqfb  37741