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

Theorem sb6 2173
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 2093 and does not require any dv condition. Theorem sb6f 2126 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
Distinct variable group:   ,

Proof of Theorem sb6
StepHypRef Expression
1 sb1 1742 . . 3
2 sb56 2172 . . 3
31, 2sylib 196 . 2
4 sb2 2093 . 2
53, 4impbii 188 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  [wsb 1739
This theorem is referenced by:  sb5  2174  2sb6  2188  sb6a  2192  2eu6  2383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-12 1854  ax-13 1999
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617  df-sb 1740
  Copyright terms: Public domain W3C validator