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Theorem elrabsf 2795
 Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2690 has implicit substitution). The hypothesis specifies that x must not be a free variable in B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
Hypothesis
Ref Expression
elrabsf.1 xB
Assertion
Ref Expression
elrabsf (A {x Bφ} ↔ (A B [A / x]φ))

Proof of Theorem elrabsf
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2760 . 2 (y = A → ([y / x]φ[A / x]φ))
2 elrabsf.1 . . 3 xB
3 nfcv 2175 . . 3 yB
4 nfv 1418 . . 3 yφ
5 nfsbc1v 2776 . . 3 x[y / x]φ
6 sbceq1a 2767 . . 3 (x = y → (φ[y / x]φ))
72, 3, 4, 5, 6cbvrab 2549 . 2 {x Bφ} = {y B[y / x]φ}
81, 7elrab2 2694 1 (A {x Bφ} ↔ (A B [A / x]φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  Ⅎwnfc 2162  {crab 2304  [wsbc 2758 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-rab 2309  df-v 2553  df-sbc 2759 This theorem is referenced by:  mpt2xopovel  5797
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