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Theorem rspc 2623
 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 2285 . 2 (x B φx(x Bφ))
2 nfcv 2156 . . . 4 xA
3 nfv 1398 . . . . 5 x A B
4 rspc.1 . . . . 5 xψ
53, 4nfim 1442 . . . 4 x(A Bψ)
6 eleq1 2078 . . . . 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 2608 . . 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 1224   = wceq 1226  Ⅎwnf 1325   ∈ wcel 1370  ∀wral 2280 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533 This theorem is referenced by:  rspcv  2625  rspc2  2634  pofun  4019  fmptcof  5252  fliftfuns  5359  qliftfuns  6097  bj-nntrans  7369
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