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Theorem sbc8g 3053
 Description: This is the closest we can get to df-sbc 3047 if we start from dfsbcq 3048 (see its comments) and dfsbcq2 3049. (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 / xφA {x φ}))

Proof of Theorem sbc8g
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3048 . 2 (y = A → ([̣y / xφ ↔ [̣A / xφ))
2 eleq1 2413 . 2 (y = A → (y {x φ} ↔ A {x φ}))
3 df-clab 2340 . . 3 (y {x φ} ↔ [y / x]φ)
4 equid 1676 . . . 4 y = y
5 dfsbcq2 3049 . . . 4 (y = y → ([y / x]φ ↔ [̣y / xφ))
64, 5ax-mp 5 . . 3 ([y / x]φ ↔ [̣y / xφ)
73, 6bitr2i 241 . 2 ([̣y / xφy {x φ})
81, 2, 7vtoclbg 2915 1 (A V → ([̣A / xφA {x φ}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  [wsb 1648   ∈ wcel 1710  {cab 2339  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-v 2861  df-sbc 3047 This theorem is referenced by: (None)
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