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Theorem bnj1096 32042
 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 1904 . . 3 (𝜒 → ∀𝑥𝜒)
3 ax-5 1904 . . 3 (𝜃 → ∀𝑥𝜃)
4 ax-5 1904 . . 3 (𝜏 → ∀𝑥𝜏)
5 bnj1096.1 . . 3 (𝜑 → ∀𝑥𝜑)
62, 3, 4, 5bnj982 32038 . 2 ((𝜒𝜃𝜏𝜑) → ∀𝑥(𝜒𝜃𝜏𝜑))
71, 6hbxfrbi 1818 1 (𝜓 → ∀𝑥𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1528   ∧ w-bnj17 31944 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 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-bnj17 31945 This theorem is referenced by:  bnj964  32203  bnj981  32210  bnj983  32211  bnj1093  32240  bnj1145  32253
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