MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb5f Structured version   Unicode version

Theorem sb5f 2184
Description: Equivalence for substitution when  y is not free in  ph. The implication "to the right" is sb1 1793 and does not require the non-freeness hypothesis. Theorem sb5 2229 replaces the non-freeness hypothesis with a dv condition. (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 2183 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
31equs45f 2148 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph ) )
42, 3bitr4i 255 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1657   F/wnf 1661   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791
This theorem is referenced by:  sb7f  2252
  Copyright terms: Public domain W3C validator