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Theorem bnj1476 32123
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 3368 . . . . . 6 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
42, 3nfcxfr 2971 . . . . 5 𝑥𝐷
54eq0f 4277 . . . 4 (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐷)
61, 5sylib 220 . . 3 (𝜓 → ∀𝑥 ¬ 𝑥𝐷)
72rabeq2i 3463 . . . . 5 (𝑥𝐷 ↔ (𝑥𝐴 ∧ ¬ 𝜑))
87notbii 322 . . . 4 𝑥𝐷 ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
9 iman 404 . . . 4 ((𝑥𝐴𝜑) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
108, 9sylbb2 240 . . 3 𝑥𝐷 → (𝑥𝐴𝜑))
116, 10sylg 1823 . 2 (𝜓 → ∀𝑥(𝑥𝐴𝜑))
1211bnj1142 32065 1 (𝜓 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wal 1535   = wceq 1537  wcel 2114  wral 3125  {crab 3129  c0 4265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rab 3134  df-dif 3912  df-nul 4266
This theorem is referenced by:  bnj1312  32334
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