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Theorem spcimdv 2936
 Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1 (φA B)
spcimdv.2 ((φ x = A) → (ψχ))
Assertion
Ref Expression
spcimdv (φ → (xψχ))
Distinct variable groups:   x,A   φ,x   χ,x
Allowed substitution hints:   ψ(x)   B(x)

Proof of Theorem spcimdv
StepHypRef Expression
1 spcimdv.2 . . . 4 ((φ x = A) → (ψχ))
21ex 423 . . 3 (φ → (x = A → (ψχ)))
32alrimiv 1631 . 2 (φx(x = A → (ψχ)))
4 spcimdv.1 . 2 (φA B)
5 nfv 1619 . . 3 xχ
6 nfcv 2489 . . 3 xA
75, 6spcimgft 2930 . 2 (x(x = A → (ψχ)) → (A B → (xψχ)))
83, 4, 7sylc 56 1 (φ → (xψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ 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:  spcdv  2937  spcimedv  2938  rspcimdv  2956
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