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Theorem sb6 2268
Description: Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" is sb2 2193 and does not require any dv condition. Theorem sb6f 2224 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
Assertion
Ref Expression
sb6  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb6
StepHypRef Expression
1 sb1 1810 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 sb56 2091 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
31, 2sylib 201 . 2  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
)
4 sb2 2193 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
53, 4impbii 192 1  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452   E.wex 1673   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  sb5  2269  2sb6  2283  sb6a  2287  2eu6  2397
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