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Theorem bnj982 30975
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 30881 . 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 2167 . . 3 ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
6 bnj982.4 . . 3 (𝜃 → ∀𝑥𝜃)
75, 6hban 2166 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) → ∀𝑥((𝜑𝜓𝜒) ∧ 𝜃))
81, 7hbxfrbi 1792 1 ((𝜑𝜓𝜒𝜃) → ∀𝑥(𝜑𝜓𝜒𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wal 1521  w-bnj17 30880
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-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-bnj17 30881
This theorem is referenced by:  bnj1096  30979  bnj1311  31218  bnj1445  31238
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