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Theorem ralidmOLD 4516
Description: Obsolete version of ralidm 4512 as of 2-Sep-2024. (Contributed by NM, 28-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ralidmOLD (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralidmOLD
StepHypRef Expression
1 rzal 4509 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑥𝐴 𝜑)
2 rzal 4509 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
31, 22thd 265 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
4 neq0 4346 . . 3 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
5 df-ral 3059 . . . . 5 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
6 nfra1 3278 . . . . . 6 𝑥𝑥𝐴 𝜑
7619.23 2200 . . . . 5 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
85, 7bitri 275 . . . 4 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
9 biimt 360 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑)))
108, 9bitr4id 290 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
114, 10sylbi 216 . 2 𝐴 = ∅ → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
123, 11pm2.61i 182 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1532   = wceq 1534  wex 1774  wcel 2099  wral 3058  c0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-ral 3059  df-dif 3950  df-nul 4324
This theorem is referenced by: (None)
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