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Theorem rspce 2645
 Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
rspc.1 xψ
rspc.2 (x = A → (φψ))
Assertion
Ref Expression
rspce ((A B ψ) → x B φ)
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem rspce
StepHypRef Expression
1 nfcv 2175 . . . 4 xA
2 nfv 1418 . . . . 5 x A B
3 rspc.1 . . . . 5 xψ
42, 3nfan 1454 . . . 4 x(A B ψ)
5 eleq1 2097 . . . . 5 (x = A → (x BA B))
6 rspc.2 . . . . 5 (x = A → (φψ))
75, 6anbi12d 442 . . . 4 (x = A → ((x B φ) ↔ (A B ψ)))
81, 4, 7spcegf 2630 . . 3 (A B → ((A B ψ) → x(x B φ)))
98anabsi5 513 . 2 ((A B ψ) → x(x B φ))
10 df-rex 2306 . 2 (x B φx(x B φ))
119, 10sylibr 137 1 ((A B ψ) → x B φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  Ⅎwnf 1346  ∃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:  rspcev  2650
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