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Theorem ralf0OLD 4474
Description: Obsolete version of ralf0 4470 as of 2-Sep-2024. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ralf0OLD.1 ¬ 𝜑
Assertion
Ref Expression
ralf0OLD (∀𝑥𝐴 𝜑𝐴 = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralf0OLD
StepHypRef Expression
1 ralf0OLD.1 . . . 4 ¬ 𝜑
2 mtt 365 . . . 4 𝜑 → (¬ 𝑥𝐴 ↔ (𝑥𝐴𝜑)))
31, 2ax-mp 5 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝜑))
43albii 1822 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝜑))
5 eq0 4302 . 2 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
6 df-ral 3064 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
74, 5, 63bitr4ri 304 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1540   = wceq 1542  wcel 2107  wral 3063  c0 4281
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 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-ral 3064  df-dif 3912  df-nul 4282
This theorem is referenced by: (None)
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