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Theorem rspcimdv 2956
 Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1 (φA B)
rspcimdv.2 ((φ x = A) → (ψχ))
Assertion
Ref Expression
rspcimdv (φ → (x B ψχ))
Distinct variable groups:   x,A   x,B   φ,x   χ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem rspcimdv
StepHypRef Expression
1 df-ral 2619 . 2 (x B ψx(x Bψ))
2 rspcimdv.1 . . 3 (φA B)
3 simpr 447 . . . . . . 7 ((φ x = A) → x = A)
43eleq1d 2419 . . . . . 6 ((φ x = A) → (x BA B))
54biimprd 214 . . . . 5 ((φ x = A) → (A Bx B))
6 rspcimdv.2 . . . . 5 ((φ x = A) → (ψχ))
75, 6imim12d 68 . . . 4 ((φ x = A) → ((x Bψ) → (A Bχ)))
82, 7spcimdv 2936 . . 3 (φ → (x(x Bψ) → (A Bχ)))
92, 8mpid 37 . 2 (φ → (x(x Bψ) → χ))
101, 9syl5bi 208 1 (φ → (x B ψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∀wral 2614 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-ral 2619  df-v 2861 This theorem is referenced by:  rspcimedv  2957  rspcdv  2958
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