Theorem nf5rd 2236

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 2234	. 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 1945  ax-6 2002  ax-7 2043  ax-12 2216

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

This theorem is used by:  spimedv  2237  alrimdd  2254  nf5diOLD  2322  hbnt  2331  hbimd  2335  dvelimhw  2378  dveeq2  2410  dveeq1  2412  axc9  2414  spimed  2420  dvelimh  2482  abidnf  3663  eusvnfb  5362  axrepnd  10718  axacndlem4  10734  bj-cbv2v  37925  bj-elgab  38067  wl-nfeqfb  38736