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Theorem bnj228 31346
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 3138 . . 3 (∀𝑥𝐴 𝜓 → (𝑥𝐴𝜓))
31, 2sylbi 209 . 2 (𝜑 → (𝑥𝐴𝜓))
43impcom 398 1 ((𝑥𝐴𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wcel 2164  wral 3117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-ral 3122
This theorem is referenced by:  bnj229  31496  bnj999  31569
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