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Theorem ralbinrald 39848
 Description: Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
Hypotheses
Ref Expression
ralbinrald.1 (𝜑𝑋𝐴)
ralbinrald.2 (𝑥𝐴𝑥 = 𝑋)
ralbinrald.3 (𝑥 = 𝑋 → (𝜓𝜃))
Assertion
Ref Expression
ralbinrald (𝜑 → (∀𝑥𝐴 𝜓𝜃))
Distinct variable groups:   𝑥,𝑋   𝑥,𝐴   𝜑,𝑥   𝜃,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem ralbinrald
StepHypRef Expression
1 ralbinrald.1 . . 3 (𝜑𝑋𝐴)
2 ralbinrald.3 . . . 4 (𝑥 = 𝑋 → (𝜓𝜃))
32adantl 481 . . 3 ((𝜑𝑥 = 𝑋) → (𝜓𝜃))
41, 3rspcdv 3285 . 2 (𝜑 → (∀𝑥𝐴 𝜓𝜃))
5 ralbinrald.2 . . . . . 6 (𝑥𝐴𝑥 = 𝑋)
62bicomd 212 . . . . . 6 (𝑥 = 𝑋 → (𝜃𝜓))
75, 6syl 17 . . . . 5 (𝑥𝐴 → (𝜃𝜓))
87adantl 481 . . . 4 ((𝜑𝑥𝐴) → (𝜃𝜓))
98biimpd 218 . . 3 ((𝜑𝑥𝐴) → (𝜃𝜓))
109ralrimdva 2952 . 2 (𝜑 → (𝜃 → ∀𝑥𝐴 𝜓))
114, 10impbid 201 1 (𝜑 → (∀𝑥𝐴 𝜓𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175 This theorem is referenced by:  dfdfat2  39860
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