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Theorem rspc 2949
 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 2619 . 2 (x B φx(x Bφ))
2 nfcv 2489 . . . 4 xA
3 nfv 1619 . . . . 5 x A B
4 rspc.1 . . . . 5 xψ
53, 4nfim 1813 . . . 4 x(A Bψ)
6 eleq1 2413 . . . . 5 (x = A → (x BA B))
7 rspc.2 . . . . 5 (x = A → (φψ))
86, 7imbi12d 311 . . . 4 (x = A → ((x Bφ) ↔ (A Bψ)))
92, 5, 8spcgf 2934 . . 3 (A B → (x(x Bφ) → (A Bψ)))
109pm2.43a 45 . 2 (A B → (x(x Bφ) → ψ))
111, 10syl5bi 208 1 (A B → (x B φψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861 This theorem is referenced by:  rspcv  2951  rspc2  2960
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