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Theorem bnj1459 32103
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1459.1 (𝜓 ↔ (𝜑𝑥𝐴))
bnj1459.2 (𝜓𝜒)
Assertion
Ref Expression
bnj1459 (𝜑 → ∀𝑥𝐴 𝜒)
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem bnj1459
StepHypRef Expression
1 bnj1459.1 . . 3 (𝜓 ↔ (𝜑𝑥𝐴))
2 bnj1459.2 . . 3 (𝜓𝜒)
31, 2sylbir 237 . 2 ((𝜑𝑥𝐴) → 𝜒)
43ralrimiva 3180 1 (𝜑 → ∀𝑥𝐴 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2107  wral 3136
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
This theorem depends on definitions:  df-bi 209  df-an 399  df-ral 3141
This theorem is referenced by:  bnj1501  32327
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