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Theorem sbc8g 2765
Description: This is the closest we can get to df-sbc 2759 if we start from dfsbcq 2760 (see its comments) and dfsbcq2 2761. (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 2760 . 2 (y = A → ([y / x]φ[A / x]φ))
2 eleq1 2097 . 2 (y = A → (y {xφ} ↔ A {xφ}))
3 df-clab 2024 . . 3 (y {xφ} ↔ [y / x]φ)
4 equid 1586 . . . 4 y = y
5 dfsbcq2 2761 . . . 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 2608 1 (A 𝑉 → ([A / x]φA {xφ}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wcel 1390  [wsb 1642  {cab 2023  [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:  bj-elssuniab  9199
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