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Theorem hban 2166
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 2064 . . 3 𝑥𝜑
3 hb.2 . . . 4 (𝜓 → ∀𝑥𝜓)
43nf5i 2064 . . 3 𝑥𝜓
52, 4nfan 1868 . 2 𝑥(𝜑𝜓)
65nf5ri 2103 1 ((𝜑𝜓) → ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by:  bnj982  30975  bnj1351  31023  bnj1352  31024  bnj1441  31037  dvelimf-o  34533  ax12indalem  34549  ax12inda2ALT  34550  hbimpg  39087  hbimpgVD  39454
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