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Theorem sbc7 3073
 Description: An equivalence for class substitution in the spirit of df-clab 2340. Note that x and A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbc7 ([̣A / xφy(y = A y / xφ))
Distinct variable groups:   y,A   φ,y   x,y
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem sbc7
StepHypRef Expression
1 sbcco 3068 . 2 ([̣A / y]̣[̣y / xφ ↔ [̣A / xφ)
2 sbc5 3070 . 2 ([̣A / y]̣[̣y / xφy(y = A y / xφ))
31, 2bitr3i 242 1 ([̣A / xφy(y = A y / xφ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642  [̣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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