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Theorem spcdv 3541
 Description: Rule of specialization, using implicit substitution. Analogous to rspcdv 3563. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
spcimdv.1 (𝜑𝐴𝐵)
spcdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
spcdv (𝜑 → (∀𝑥𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcdv
StepHypRef Expression
1 spcimdv.1 . 2 (𝜑𝐴𝐵)
2 spcdv.2 . . 3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
32biimpd 232 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
41, 3spcimdv 3540 1 (𝜑 → (∀𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870 This theorem is referenced by:  spcgv  3543  mrissmrcd  16906
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