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Theorem hban 2275
 Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (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 2173 . . 3 𝑥𝜑
3 hb.2 . . . 4 (𝜓 → ∀𝑥𝜓)
43nf5i 2173 . . 3 𝑥𝜓
52, 4nfan 1977 . 2 𝑥(𝜑𝜓)
65nf5ri 2212 1 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859 This theorem is referenced by:  bnj982  31156  bnj1351  31204  bnj1352  31205  bnj1441  31218  dvelimf-o  34718  ax12indalem  34734  ax12inda2ALT  34735  hbimpg  39272  hbimpgVD  39639
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