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Theorem ralidmOLD 4402
Description: Obsolete version of ralidm 4398 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 4395 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑥𝐴 𝜑)
2 rzal 4395 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
31, 22thd 268 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
4 neq0 4234 . . 3 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
5 df-ral 3058 . . . . 5 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
6 nfra1 3131 . . . . . 6 𝑥𝑥𝐴 𝜑
7619.23 2213 . . . . 5 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
85, 7bitri 278 . . . 4 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
9 biimt 364 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑)))
108, 9bitr4id 293 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
114, 10sylbi 220 . 2 𝐴 = ∅ → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
123, 11pm2.61i 185 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1540   = wceq 1542  wex 1786  wcel 2114  wral 3053  c0 4211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-ral 3058  df-dif 3846  df-nul 4212
This theorem is referenced by: (None)
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