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Theorem sb5rf 2189
Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.)
Hypothesis
Ref Expression
sb5rf.1  |-  F/ y
ph
Assertion
Ref Expression
sb5rf  |-  ( ph  <->  E. y ( y  =  x  /\  [ y  /  x ] ph ) )

Proof of Theorem sb5rf
StepHypRef Expression
1 sb5rf.1 . . 3  |-  F/ y
ph
2 sbequ12r 2021 . . 3  |-  ( y  =  x  ->  ( [ y  /  x ] ph  <->  ph ) )
31, 2equsex 2064 . 2  |-  ( E. y ( y  =  x  /\  [ y  /  x ] ph ) 
<-> 
ph )
43bicomi 202 1  |-  ( ph  <->  E. y ( y  =  x  /\  [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367   E.wex 1633   F/wnf 1637   [wsb 1763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-12 1878  ax-13 2026
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638  df-sb 1764
This theorem is referenced by: (None)
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