Theorem nf5rd 2232

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 2230	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  spimedv  2233  alrimdd  2250  nf5di  2318  hbnt  2327  hbimd  2331  dvelimhw  2374  dveeq2  2407  dveeq1  2409  axc9  2411  spimed  2417  dvelimh  2479  abidnf  3660  eusvnfb  5358  axrepnd  10604  axacndlem4  10620  bj-cbv2v  37542  bj-elgab  37684  wl-nfeqfb  38300