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Theorem ralbinrald 41715
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 467 . . 3 ((𝜑𝑥 = 𝑋) → (𝜓𝜃))
41, 3rspcdv 3463 . 2 (𝜑 → (∀𝑥𝐴 𝜓𝜃))
5 ralbinrald.2 . . . . . 6 (𝑥𝐴𝑥 = 𝑋)
62bicomd 213 . . . . . 6 (𝑥 = 𝑋 → (𝜃𝜓))
75, 6syl 17 . . . . 5 (𝑥𝐴 → (𝜃𝜓))
87adantl 467 . . . 4 ((𝜑𝑥𝐴) → (𝜃𝜓))
98biimpd 219 . . 3 ((𝜑𝑥𝐴) → (𝜃𝜓))
109ralrimdva 3118 . 2 (𝜑 → (𝜃 → ∀𝑥𝐴 𝜓))
114, 10impbid 202 1 (𝜑 → (∀𝑥𝐴 𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353
This theorem is referenced by:  dfdfat2  41727
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