Theorem nf5rd 2204

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 2202	. 2 ⊢ (Ⅎ𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-ex 1782  df-nf 1786

This theorem is referenced by:  spimedv  2205  alrimdd  2222  nf5di  2292  hbnt  2301  hbimd  2305  dvelimhw  2350  dveeq2  2383  dveeq1  2385  axc9  2387  spimed  2393  dvelimh  2455  abidnf  3662  eusvnfb  5340  axrepnd  10517  axacndlem4  10533  bj-cbv2v  37046  bj-elgab  37187  wl-nfeqfb  37791