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Theorem rspcimedv 2652
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 . . 3 (φA B)
2 simpr 103 . . . . . . 7 ((φ x = A) → x = A)
32eleq1d 2103 . . . . . 6 ((φ x = A) → (x BA B))
43biimprd 147 . . . . 5 ((φ x = A) → (A Bx B))
5 rspcimedv.2 . . . . 5 ((φ x = A) → (χψ))
64, 5anim12d 318 . . . 4 ((φ x = A) → ((A B χ) → (x B ψ)))
71, 6spcimedv 2633 . . 3 (φ → ((A B χ) → x(x B ψ)))
81, 7mpand 405 . 2 (φ → (χx(x B ψ)))
9 df-rex 2306 . 2 (x B ψx(x B ψ))
108, 9syl6ibr 151 1 (φ → (χx B ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   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:  rspcedv  2654
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