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Theorem ralbinrald 42177
 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 475 . . 3 ((𝜑𝑥 = 𝑋) → (𝜓𝜃))
41, 3rspcdv 3514 . 2 (𝜑 → (∀𝑥𝐴 𝜓𝜃))
5 ralbinrald.2 . . . . . 6 (𝑥𝐴𝑥 = 𝑋)
62bicomd 215 . . . . . 6 (𝑥 = 𝑋 → (𝜃𝜓))
75, 6syl 17 . . . . 5 (𝑥𝐴 → (𝜃𝜓))
87adantl 475 . . . 4 ((𝜑𝑥𝐴) → (𝜃𝜓))
98biimpd 221 . . 3 ((𝜑𝑥𝐴) → (𝜃𝜓))
109ralrimdva 3151 . 2 (𝜑 → (𝜃 → ∀𝑥𝐴 𝜓))
114, 10impbid 204 1 (𝜑 → (∀𝑥𝐴 𝜓𝜃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-v 3400 This theorem is referenced by:  dfdfat2  42183
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