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Theorem rspcedv 2654
 Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcdv.1 (φA B)
rspcdv.2 ((φ x = A) → (ψχ))
Assertion
Ref Expression
rspcedv (φ → (χx B ψ))
Distinct variable groups:   x,A   x,B   φ,x   χ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem rspcedv
StepHypRef Expression
1 rspcdv.1 . 2 (φA B)
2 rspcdv.2 . . 3 ((φ x = A) → (ψχ))
32biimprd 147 . 2 ((φ x = A) → (χψ))
41, 3rspcimedv 2652 1 (φ → (χx B ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃wrex 2301 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-rex 2306  df-v 2553 This theorem is referenced by:  rexxfrd  4161  ltexnqq  6391  halfnqq  6393  ltbtwnnqq  6398  genpml  6500  genpmu  6501  genprndl  6504  genprndu  6505  axarch  6773  apreap  7371
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