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Theorem bnj1276 32780
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 2298 . 2 ((𝜑𝜓𝜒) → ∀𝑥(𝜑𝜓𝜒))
61, 5hbxfrbi 1827 1 (𝜃 → ∀𝑥𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1086  wal 1537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787
This theorem is referenced by: (None)
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