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Theorem bnj1276 32090
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1276.1 (𝜑 → ∀𝑥𝜑)
bnj1276.2 (𝜓 → ∀𝑥𝜓)
bnj1276.3 (𝜒 → ∀𝑥𝜒)
bnj1276.4 (𝜃 ↔ (𝜑𝜓𝜒))
Assertion
Ref Expression
bnj1276 (𝜃 → ∀𝑥𝜃)

Proof of Theorem bnj1276
StepHypRef Expression
1 bnj1276.4 . 2 (𝜃 ↔ (𝜑𝜓𝜒))
2 bnj1276.1 . . 3 (𝜑 → ∀𝑥𝜑)
3 bnj1276.2 . . 3 (𝜓 → ∀𝑥𝜓)
4 bnj1276.3 . . 3 (𝜒 → ∀𝑥𝜒)
52, 3, 4hb3an 2308 . 2 ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
61, 5hbxfrbi 1824 1 (𝜃 → ∀𝑥𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083  ∀wal 1534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784 This theorem is referenced by: (None)
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