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Theorem ralidm 3514
Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
Assertion
Ref Expression
ralidm (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralidm
StepHypRef Expression
1 nfra1 2501 . . 3 𝑥𝑥𝐴𝑥𝐴 𝜑
2 anidm 394 . . . 4 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
3 rsp2 2520 . . . 4 (∀𝑥𝐴𝑥𝐴 𝜑 → ((𝑥𝐴𝑥𝐴) → 𝜑))
42, 3syl5bir 152 . . 3 (∀𝑥𝐴𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
51, 4ralrimi 2541 . 2 (∀𝑥𝐴𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜑)
6 ax-1 6 . . . 4 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
7 nfra1 2501 . . . . 5 𝑥𝑥𝐴 𝜑
8719.23 1671 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
96, 8sylibr 133 . . 3 (∀𝑥𝐴 𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
10 df-ral 2453 . . 3 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
119, 10sylibr 133 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴𝑥𝐴 𝜑)
125, 11impbii 125 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346  wex 1485  wcel 2141  wral 2448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453
This theorem is referenced by:  issref  4991  cnvpom  5151
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