Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  hb3an Structured version   Visualization version   GIF version

Theorem hb3an 2276
 Description: If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (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 2173 . . 3 𝑥𝜑
3 hb.2 . . . 4 (𝜓 → ∀𝑥𝜓)
43nf5i 2173 . . 3 𝑥𝜓
5 hb.3 . . . 4 (𝜒 → ∀𝑥𝜒)
65nf5i 2173 . . 3 𝑥𝜒
72, 4, 6nf3an 1980 . 2 𝑥(𝜑𝜓𝜒)
87nf5ri 2212 1 ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1072  ∀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-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859 This theorem is referenced by:  bnj982  31177  bnj1276  31213  bnj1350  31224
 Copyright terms: Public domain W3C validator