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Theorem bnj228 31110
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 3067 . . 3 (∀𝑥𝐴 𝜓 → (𝑥𝐴𝜓))
31, 2sylbi 207 . 2 (𝜑 → (𝑥𝐴𝜓))
43impcom 445 1 ((𝑥𝐴𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-ral 3055
This theorem is referenced by:  bnj229  31261  bnj999  31334
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