Theorem nf5rd 2197

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 2195	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

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:  spimedv  2198  alrimdd  2215  nf5di  2286  hbnt  2295  hbimd  2299  dvelimhw  2347  dveeq2  2383  dveeq1  2385  axc9  2387  spimed  2393  dvelimh  2455  abidnf  3690  eusvnfb  5368  axrepnd  10613  axacndlem4  10629  bj-cbv2v  36821  bj-elgab  36962  wl-nfeqfb  37559