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Theorem sbc7 3132
Description: An equivalence for class substitution in the spirit of df-clab 2375. 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  /  x ]. ph  <->  E. y
( y  =  A  /\  [. y  /  x ]. ph ) )
Distinct variable groups:    y, A    ph, y    x, y
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem sbc7
StepHypRef Expression
1 sbcco 3127 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
2 sbc5 3129 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  E. y
( y  =  A  /\  [. y  /  x ]. ph ) )
31, 2bitr3i 243 1  |-  ( [. A  /  x ]. ph  <->  E. y
( y  =  A  /\  [. y  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649   [.wsbc 3105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-sbc 3106
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