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Theorem rspcdv 2958
 Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcdv.1 (φA B)
rspcdv.2 ((φ x = A) → (ψχ))
Assertion
Ref Expression
rspcdv (φ → (x B ψχ))
Distinct variable groups:   x,A   x,B   φ,x   χ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem rspcdv
StepHypRef Expression
1 rspcdv.1 . 2 (φA B)
2 rspcdv.2 . . 3 ((φ x = A) → (ψχ))
32biimpd 198 . 2 ((φ x = A) → (ψχ))
41, 3rspcimdv 2956 1 (φ → (x B ψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = 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: (None)
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