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Theorem bnj982 31955
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 31862 . 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 2303 . . 3 ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
6 bnj982.4 . . 3 (𝜃 → ∀𝑥𝜃)
75, 6hban 2302 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) → ∀𝑥((𝜑𝜓𝜒) ∧ 𝜃))
81, 7hbxfrbi 1818 1 ((𝜑𝜓𝜒𝜃) → ∀𝑥(𝜑𝜓𝜒𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081  wal 1528  w-bnj17 31861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-bnj17 31862
This theorem is referenced by:  bnj1096  31959  bnj1311  32199  bnj1445  32219
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