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Theorem rspcimdv 2651
 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 2305 . 2 (x B ψx(x Bψ))
2 rspcimdv.1 . . 3 (φA B)
3 simpr 103 . . . . . . 7 ((φ x = A) → x = A)
43eleq1d 2103 . . . . . 6 ((φ x = A) → (x BA B))
54biimprd 147 . . . . 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 2631 . . 3 (φ → (x(x Bψ) → (A Bχ)))
92, 8mpid 37 . 2 (φ → (x(x Bψ) → χ))
101, 9syl5bi 141 1 (φ → (x B ψχ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390  ∀wral 2300 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553 This theorem is referenced by:  rspcdv  2653
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