Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj982 Structured version   Visualization version   GIF version

Theorem bnj982 31318
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj982.1 (𝜑 → ∀𝑥𝜑)
bnj982.2 (𝜓 → ∀𝑥𝜓)
bnj982.3 (𝜒 → ∀𝑥𝜒)
bnj982.4 (𝜃 → ∀𝑥𝜃)
Assertion
Ref Expression
bnj982 ((𝜑𝜓𝜒𝜃) → ∀𝑥(𝜑𝜓𝜒𝜃))

Proof of Theorem bnj982
StepHypRef Expression
1 df-bnj17 31225 . 2 ((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜒) ∧ 𝜃))
2 bnj982.1 . . . 4 (𝜑 → ∀𝑥𝜑)
3 bnj982.2 . . . 4 (𝜓 → ∀𝑥𝜓)
4 bnj982.3 . . . 4 (𝜒 → ∀𝑥𝜒)
52, 3, 4hb3an 2306 . . 3 ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
6 bnj982.4 . . 3 (𝜃 → ∀𝑥𝜃)
75, 6hban 2305 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) → ∀𝑥((𝜑𝜓𝜒) ∧ 𝜃))
81, 7hbxfrbi 1919 1 ((𝜑𝜓𝜒𝜃) → ∀𝑥(𝜑𝜓𝜒𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107  wal 1650  w-bnj17 31224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-bnj17 31225
This theorem is referenced by:  bnj1096  31322  bnj1311  31561  bnj1445  31581
  Copyright terms: Public domain W3C validator