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Theorem bnj228 31904
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj228.1 (𝜑 ↔ ∀𝑥𝐴 𝜓)
Assertion
Ref Expression
bnj228 ((𝑥𝐴𝜑) → 𝜓)

Proof of Theorem bnj228
StepHypRef Expression
1 bnj228.1 . . 3 (𝜑 ↔ ∀𝑥𝐴 𝜓)
2 rsp 3202 . . 3 (∀𝑥𝐴 𝜓 → (𝑥𝐴𝜓))
31, 2sylbi 218 . 2 (𝜑 → (𝑥𝐴𝜓))
43impcom 408 1 ((𝑥𝐴𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-ral 3140
This theorem is referenced by:  bnj229  32055  bnj999  32128
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