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Theorem sbc7 3018
Description: An equivalence for class substitution in the spirit of df-clab 2270. 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 3013 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
2 sbc5 3015 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  E. y
( y  =  A  /\  [. y  /  x ]. ph ) )
31, 2bitr3i 242 1  |-  ( [. A  /  x ]. ph  <->  E. y
( y  =  A  /\  [. y  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623   [.wsbc 2991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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