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Theorem bnj1476 30617
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 3116 . . . . . 6 𝑥{𝑥𝐴 ∣ ¬ 𝜑}
42, 3nfcxfr 2765 . . . . 5 𝑥𝐷
54eq0f 3906 . . . 4 (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐷)
61, 5sylib 208 . . 3 (𝜓 → ∀𝑥 ¬ 𝑥𝐷)
72rabeq2i 3188 . . . . 5 (𝑥𝐷 ↔ (𝑥𝐴 ∧ ¬ 𝜑))
87notbii 310 . . . 4 𝑥𝐷 ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
9 iman 440 . . . 4 ((𝑥𝐴𝜑) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝜑))
108, 9sylbb2 228 . . 3 𝑥𝐷 → (𝑥𝐴𝜑))
116, 10sylg 1747 . 2 (𝜓 → ∀𝑥(𝑥𝐴𝜑))
1211bnj1142 30560 1 (𝜓 → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1992  wral 2912  {crab 2916  c0 3896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-nul 3897
This theorem is referenced by:  bnj1312  30826
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