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Theorem sb5f 2125
Description: Equivalence for substitution when  y is not free in  ph. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sb6f.1  |-  F/ y
ph
Assertion
Ref Expression
sb5f  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )

Proof of Theorem sb5f
StepHypRef Expression
1 sb6f.1 . . 3  |-  F/ y
ph
21sb6f 2124 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
31equs45f 2089 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
42, 3bitr4i 245 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551   F/wnf 1554   [wsb 1659
This theorem is referenced by:  sb7f  2198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-sb 1660
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