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Theorem ralidmw 4464
Description: Idempotent law for restricted quantifier. Weak version of ralidm 4468, which does not require ax-10 2138, ax-12 2172, but requires ax-8 2109. (Contributed by Gino Giotto, 30-Sep-2024.)
Hypothesis
Ref Expression
ralidmw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
ralidmw (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem ralidmw
StepHypRef Expression
1 df-ral 3064 . . . . 5 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
21imbi2i 336 . . . 4 ((𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
32albii 1822 . . 3 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥(𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
4 pm2.21 123 . . . . . 6 𝑥𝐴 → (𝑥𝐴𝜑))
5 eleq1w 2821 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
6 ralidmw.1 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
75, 6imbi12d 345 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜓)))
87spw 2038 . . . . . 6 (∀𝑥(𝑥𝐴𝜑) → (𝑥𝐴𝜑))
94, 8ja 186 . . . . 5 ((𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → (𝑥𝐴𝜑))
109alimi 1814 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) → ∀𝑥(𝑥𝐴𝜑))
117hba1w 2051 . . . . 5 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥𝑥(𝑥𝐴𝜑))
12 ax-1 6 . . . . 5 (∀𝑥(𝑥𝐴𝜑) → (𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
1311, 12alrimih 1827 . . . 4 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)))
1410, 13impbii 208 . . 3 (∀𝑥(𝑥𝐴 → ∀𝑥(𝑥𝐴𝜑)) ↔ ∀𝑥(𝑥𝐴𝜑))
153, 14bitri 275 . 2 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥(𝑥𝐴𝜑))
16 df-ral 3064 . 2 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
1715, 16, 13bitr4i 303 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wcel 2107  wral 3063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2816  df-ral 3064
This theorem is referenced by:  dfwe2  7701
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