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Theorem rspc 2644
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
rspc.1 xψ
rspc.2 (x = A → (φψ))
Assertion
Ref Expression
rspc (A B → (x B φψ))
Distinct variable groups:   x,A   x,B
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem rspc
StepHypRef Expression
1 df-ral 2305 . 2 (x B φx(x Bφ))
2 nfcv 2175 . . . 4 xA
3 nfv 1418 . . . . 5 x A B
4 rspc.1 . . . . 5 xψ
53, 4nfim 1461 . . . 4 x(A Bψ)
6 eleq1 2097 . . . . 5 (x = A → (x BA B))
7 rspc.2 . . . . 5 (x = A → (φψ))
86, 7imbi12d 223 . . . 4 (x = A → ((x Bφ) ↔ (A Bψ)))
92, 5, 8spcgf 2629 . . 3 (A B → (x(x Bφ) → (A Bψ)))
109pm2.43a 45 . 2 (A B → (x(x Bφ) → ψ))
111, 10syl5bi 141 1 (A B → (x B φψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   = wceq 1242  wnf 1346   wcel 1390  wral 2300
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-ral 2305  df-v 2553
This theorem is referenced by:  rspcv  2646  rspc2  2655  pofun  4040  fmptcof  5274  fliftfuns  5381  qliftfuns  6126  bj-nntrans  9385
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