Theorem nf5rd 2235

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 2233	. 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 2216

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

This theorem is used by:  spimedv  2236  alrimdd  2253  nf5di  2322  hbnt  2331  hbimd  2335  dvelimhw  2379  dveeq2  2412  dveeq1  2414  axc9  2416  spimed  2422  dvelimh  2484  abidnf  3667  eusvnfb  5366  axrepnd  10596  axacndlem4  10612  bj-cbv2v  37492  bj-elgab  37634  wl-nfeqfb  38250