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Theorem ralf0OLD 4051
 Description: Obsolete proof of ralf0 4050 as of 14-Jul-2021. (Contributed by NM, 26-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
ralf0.1 ¬ 𝜑
Assertion
Ref Expression
ralf0OLD (∀𝑥𝐴 𝜑𝐴 = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralf0OLD
StepHypRef Expression
1 ralf0.1 . . . . 5 ¬ 𝜑
2 con3 149 . . . . 5 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
31, 2mpi 20 . . . 4 ((𝑥𝐴𝜑) → ¬ 𝑥𝐴)
43alimi 1736 . . 3 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥 ¬ 𝑥𝐴)
5 df-ral 2912 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 eq0 3905 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
74, 5, 63imtr4i 281 . 2 (∀𝑥𝐴 𝜑𝐴 = ∅)
8 rzal 4045 . 2 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
97, 8impbii 199 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1478   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∅c0 3891 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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 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 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3558  df-nul 3892 This theorem is referenced by: (None)
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