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Theorem rspcdvinvd 38791
Description: If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypotheses
Ref Expression
rspcdvinvd.1 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
rspcdvinvd.2 (𝜑𝐴𝐵)
rspcdvinvd.3 (𝜑 → ∀𝑥𝐵 𝜓)
Assertion
Ref Expression
rspcdvinvd (𝜑𝜒)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdvinvd
StepHypRef Expression
1 rspcdvinvd.3 . 2 (𝜑 → ∀𝑥𝐵 𝜓)
2 rspcdvinvd.2 . . 3 (𝜑𝐴𝐵)
3 rspcdvinvd.1 . . 3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
42, 3rspcdv 3343 . 2 (𝜑 → (∀𝑥𝐵 𝜓𝜒))
51, 4mpd 15 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233
This theorem is referenced by:  imo72b2  38792
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