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Theorem hban 2303
Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Hypotheses
Ref Expression
hb.1 (𝜑 → ∀𝑥𝜑)
hb.2 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
hban ((𝜑𝜓) → ∀𝑥(𝜑𝜓))

Proof of Theorem hban
StepHypRef Expression
1 hb.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21nf5i 2189 . . 3 𝑥𝜑
3 hb.2 . . . 4 (𝜓 → ∀𝑥𝜓)
43nf5i 2189 . . 3 𝑥𝜓
52, 4nfan 1990 . 2 𝑥(𝜑𝜓)
65nf5ri 2229 1 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-10 2184  ax-12 2213
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864
This theorem is referenced by:  bnj982  31166  bnj1351  31214  bnj1352  31215  bnj1441  31228  dvelimf-o  34702  ax12indalem  34718  ax12inda2ALT  34719  hbimpg  39262  hbimpgVD  39628
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