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Theorem bnj1476 30171
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 3099 . . . . . 6 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
42, 3nfcxfr 2749 . . . . 5 𝑥𝐷
54eq0f 3884 . . . 4 (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐷)
61, 5sylib 207 . . 3 (𝜓 → ∀𝑥 ¬ 𝑥𝐷)
72rabeq2i 3170 . . . . . 6 (𝑥𝐷 ↔ (𝑥𝐴 ∧ ¬ 𝜑))
87notbii 309 . . . . 5 𝑥𝐷 ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
9 iman 439 . . . . 5 ((𝑥𝐴𝜑) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
108, 9sylbb2 227 . . . 4 𝑥𝐷 → (𝑥𝐴𝜑))
1110alimi 1730 . . 3 (∀𝑥 ¬ 𝑥𝐷 → ∀𝑥(𝑥𝐴𝜑))
126, 11syl 17 . 2 (𝜓 → ∀𝑥(𝑥𝐴𝜑))
1312bnj1142 30114 1 (𝜓 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1473   = wceq 1475  wcel 1977  wral 2896  {crab 2900  c0 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-nul 3875
This theorem is referenced by:  bnj1312  30380
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