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Theorem bnj1476 34840
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1476.1 𝐷 = {𝑥𝐴 ∣ ¬ 𝜑}
bnj1476.2 (𝜓𝐷 = ∅)
Assertion
Ref Expression
bnj1476 (𝜓 → ∀𝑥𝐴 𝜑)

Proof of Theorem bnj1476
StepHypRef Expression
1 bnj1476.2 . . . 4 (𝜓𝐷 = ∅)
2 bnj1476.1 . . . . . 6 𝐷 = {𝑥𝐴 ∣ ¬ 𝜑}
3 nfrab1 3454 . . . . . 6 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
42, 3nfcxfr 2901 . . . . 5 𝑥𝐷
54eq0f 4353 . . . 4 (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐷)
61, 5sylib 218 . . 3 (𝜓 → ∀𝑥 ¬ 𝑥𝐷)
72reqabi 3457 . . . . 5 (𝑥𝐷 ↔ (𝑥𝐴 ∧ ¬ 𝜑))
87notbii 320 . . . 4 𝑥𝐷 ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
9 iman 401 . . . 4 ((𝑥𝐴𝜑) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
108, 9sylbb2 238 . . 3 𝑥𝐷 → (𝑥𝐴𝜑))
116, 10sylg 1820 . 2 (𝜓 → ∀𝑥(𝑥𝐴𝜑))
1211bnj1142 34782 1 (𝜓 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  wral 3059  {crab 3433  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-dif 3966  df-nul 4340
This theorem is referenced by:  bnj1312  35051
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