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Theorem hb3an 2307
Description: If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Hypotheses
Ref Expression
hb.1 (𝜑 → ∀𝑥𝜑)
hb.2 (𝜓 → ∀𝑥𝜓)
hb.3 (𝜒 → ∀𝑥𝜒)
Assertion
Ref Expression
hb3an ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))

Proof of Theorem hb3an
StepHypRef Expression
1 hb.1 . . . 4 (𝜑 → ∀𝑥𝜑)
21nf5i 2191 . . 3 𝑥𝜑
3 hb.2 . . . 4 (𝜓 → ∀𝑥𝜓)
43nf5i 2191 . . 3 𝑥𝜓
5 hb.3 . . . 4 (𝜒 → ∀𝑥𝜒)
65nf5i 2191 . . 3 𝑥𝜒
72, 4, 6nf3an 1993 . 2 𝑥(𝜑𝜓𝜒)
87nf5ri 2231 1 ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1100  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 2069  ax-7 2105  ax-10 2186  ax-12 2215
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864
This theorem is referenced by:  bnj982  31177  bnj1276  31213  bnj1350  31224
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