MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb6 Unicode version

Theorem sb6 1993
Description: Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.)
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 sb56 1992 . . 3  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
21anbi2i 678 . 2  |-  ( ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)  <->  ( ( x  =  y  ->  ph )  /\  A. x ( x  =  y  ->  ph )
) )
3 df-sb 1884 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
4 ax-4 1692 . . 3  |-  ( A. x ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ph ) )
54pm4.71ri 617 . 2  |-  ( A. x ( x  =  y  ->  ph )  <->  ( (
x  =  y  ->  ph )  /\  A. x
( x  =  y  ->  ph ) ) )
62, 3, 53bitr4i 270 1  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537   [wsb 1883
This theorem is referenced by:  sb5  1994  2sb6  2076  sb6a  2079  exsb  2093  sbal2  2103
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
  Copyright terms: Public domain W3C validator