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Theorem spcdv 2937
 Description: Rule of specialization, using implicit substitution. Analogous to rspcdv 2958. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
spcimdv.1 (φA B)
spcdv.2 ((φ x = 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 (φA B)
2 spcdv.2 . . 3 ((φ x = A) → (ψχ))
32biimpd 198 . 2 ((φ x = A) → (ψχ))
41, 3spcimdv 2936 1 (φ → (xψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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