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Theorem sbc6g 2782
 Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
sbc6g (A 𝑉 → ([A / x]φx(x = Aφ)))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem sbc6g
StepHypRef Expression
1 nfe1 1382 . . 3 xx(x = A φ)
2 ceqex 2665 . . 3 (x = A → (φx(x = A φ)))
31, 2ceqsalg 2576 . 2 (A 𝑉 → (x(x = Aφ) ↔ x(x = A φ)))
4 sbc5 2781 . 2 ([A / x]φx(x = A φ))
53, 4syl6rbbr 188 1 (A 𝑉 → ([A / x]φx(x = Aφ)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  [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-v 2553  df-sbc 2759 This theorem is referenced by:  sbc6  2783  sbciegft  2787  ralsns  3399  fz1sbc  8708
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