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Theorem rspce 2950
 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 2489 . . . 4 xA
2 nfv 1619 . . . . 5 x A B
3 rspc.1 . . . . 5 xψ
42, 3nfan 1824 . . . 4 x(A B ψ)
5 eleq1 2413 . . . . 5 (x = A → (x BA B))
6 rspc.2 . . . . 5 (x = A → (φψ))
75, 6anbi12d 691 . . . 4 (x = A → ((x B φ) ↔ (A B ψ)))
81, 4, 7spcegf 2935 . . 3 (A B → ((A B ψ) → x(x B φ)))
98anabsi5 790 . 2 ((A B ψ) → x(x B φ))
10 df-rex 2620 . 2 (x B φx(x B φ))
119, 10sylibr 203 1 ((A B ψ) → x B φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-rex 2620  df-v 2861 This theorem is referenced by:  rspcev  2955
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