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Theorem rspcimedv 2957
 Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1 (φA B)
rspcimedv.2 ((φ x = A) → (χψ))
Assertion
Ref Expression
rspcimedv (φ → (χx B ψ))
Distinct variable groups:   x,A   x,B   φ,x   χ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem rspcimedv
StepHypRef Expression
1 rspcimdv.1 . . . 4 (φA B)
2 rspcimedv.2 . . . . 5 ((φ x = A) → (χψ))
32con3d 125 . . . 4 ((φ x = A) → (¬ ψ → ¬ χ))
41, 3rspcimdv 2956 . . 3 (φ → (x B ¬ ψ → ¬ χ))
54con2d 107 . 2 (φ → (χ → ¬ x B ¬ ψ))
6 dfrex2 2627 . 2 (x B ψ ↔ ¬ x B ¬ ψ)
75, 6syl6ibr 218 1 (φ → (χx B ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-rex 2620  df-v 2861 This theorem is referenced by:  rspcedv  2959
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