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Theorem ralbinrald 39755
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 480 . . 3 ((𝜑𝑥 = 𝑋) → (𝜓𝜃))
41, 3rspcdv 3189 . 2 (𝜑 → (∀𝑥𝐴 𝜓𝜃))
5 ralbinrald.2 . . . . . 6 (𝑥𝐴𝑥 = 𝑋)
62bicomd 211 . . . . . 6 (𝑥 = 𝑋 → (𝜃𝜓))
75, 6syl 17 . . . . 5 (𝑥𝐴 → (𝜃𝜓))
87adantl 480 . . . 4 ((𝜑𝑥𝐴) → (𝜃𝜓))
98biimpd 217 . . 3 ((𝜑𝑥𝐴) → (𝜃𝜓))
109ralrimdva 2856 . 2 (𝜑 → (𝜃 → ∀𝑥𝐴 𝜓))
114, 10impbid 200 1 (𝜑 → (∀𝑥𝐴 𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wral 2800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-v 3079
This theorem is referenced by:  dfdfat2  39768
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