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Theorem sbc8g 2748
 Description: This is the closest we can get to df-sbc 2742 if we start from dfsbcq 2743 (see its comments) and dfsbcq2 2744. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbc8g (A 𝑉 → ([A / x]φA {xφ}))

Proof of Theorem sbc8g
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2743 . 2 (y = A → ([y / x]φ[A / x]φ))
2 eleq1 2082 . 2 (y = A → (y {xφ} ↔ A {xφ}))
3 df-clab 2009 . . 3 (y {xφ} ↔ [y / x]φ)
4 equid 1571 . . . 4 y = y
5 dfsbcq2 2744 . . . 4 (y = y → ([y / x]φ[y / x]φ))
64, 5ax-mp 7 . . 3 ([y / x]φ[y / x]φ)
73, 6bitr2i 174 . 2 ([y / x]φy {xφ})
81, 2, 7vtoclbg 2591 1 (A 𝑉 → ([A / x]φA {xφ}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1374  [wsb 1627  {cab 2008  [wsbc 2741 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742 This theorem is referenced by:  bj-elssuniab  7037
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