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Theorem bnj1476 32814
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 3316 . . . . . 6 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
42, 3nfcxfr 2905 . . . . 5 𝑥𝐷
54eq0f 4276 . . . 4 (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐷)
61, 5sylib 217 . . 3 (𝜓 → ∀𝑥 ¬ 𝑥𝐷)
72rabeq2i 3421 . . . . 5 (𝑥𝐷 ↔ (𝑥𝐴 ∧ ¬ 𝜑))
87notbii 320 . . . 4 𝑥𝐷 ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
9 iman 402 . . . 4 ((𝑥𝐴𝜑) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
108, 9sylbb2 237 . . 3 𝑥𝐷 → (𝑥𝐴𝜑))
116, 10sylg 1825 . 2 (𝜓 → ∀𝑥(𝑥𝐴𝜑))
1211bnj1142 32756 1 (𝜓 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1537   = wceq 1539  wcel 2106  wral 3064  {crab 3068  c0 4258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-dif 3891  df-nul 4259
This theorem is referenced by:  bnj1312  33025
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