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Theorem bnj228 29850
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 2912 . . 3 (∀𝑥𝐴 𝜓 → (𝑥𝐴𝜓))
31, 2sylbi 205 . 2 (𝜑 → (𝑥𝐴𝜓))
43impcom 444 1 ((𝑥𝐴𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  wral 2895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-ral 2900
This theorem is referenced by:  bnj229  30001  bnj999  30074
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