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Theorem bnj1096 30827
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1096.1 (𝜑 → ∀𝑥𝜑)
bnj1096.2 (𝜓 ↔ (𝜒𝜃𝜏𝜑))
Assertion
Ref Expression
bnj1096 (𝜓 → ∀𝑥𝜓)
Distinct variable groups:   𝜒,𝑥   𝜏,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bnj1096
StepHypRef Expression
1 bnj1096.2 . 2 (𝜓 ↔ (𝜒𝜃𝜏𝜑))
2 ax-5 1837 . . 3 (𝜒 → ∀𝑥𝜒)
3 ax-5 1837 . . 3 (𝜃 → ∀𝑥𝜃)
4 ax-5 1837 . . 3 (𝜏 → ∀𝑥𝜏)
5 bnj1096.1 . . 3 (𝜑 → ∀𝑥𝜑)
62, 3, 4, 5bnj982 30823 . 2 ((𝜒𝜃𝜏𝜑) → ∀𝑥(𝜒𝜃𝜏𝜑))
71, 6hbxfrbi 1750 1 (𝜓 → ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479  w-bnj17 30726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-bnj17 30727
This theorem is referenced by:  bnj964  30987  bnj981  30994  bnj983  30995  bnj1093  31022  bnj1145  31035
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