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Theorem ralidm 4518
Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) Reduce axiom usage. (Revised by GG, 2-Sep-2024.)
Assertion
Ref Expression
ralidm (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)

Proof of Theorem ralidm
StepHypRef Expression
1 df-ral 3060 . 2 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
2 df-ral 3060 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
3 ax-1 6 . . . . 5 (∀𝑥(𝑥𝐴𝜑) → (𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
43axc4i 2321 . . . 4 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
5 pm2.21 123 . . . . . 6 𝑥𝐴 → (𝑥𝐴𝜑))
6 sp 2181 . . . . . 6 (∀𝑥(𝑥𝐴𝜑) → (𝑥𝐴𝜑))
75, 6ja 186 . . . . 5 ((𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → (𝑥𝐴𝜑))
87alimi 1808 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴𝜑))
94, 8impbii 209 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
102bicomi 224 . . . . 5 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑥𝐴 𝜑)
1110imbi2i 336 . . . 4 ((𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) ↔ (𝑥𝐴 → ∀𝑥𝐴 𝜑))
1211albii 1816 . . 3 (∀𝑥(𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
132, 9, 123bitrri 298 . 2 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
141, 13bitri 275 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535  wcel 2106  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1777  df-nf 1781  df-ral 3060
This theorem is referenced by:  cnvpo  6309
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