Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb5 Structured version   GIF version

Theorem sb5 1764
 Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
Assertion
Ref Expression
sb5 ([y / x]φx(x = y φ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem sb5
StepHypRef Expression
1 sb6 1763 . 2 ([y / x]φx(x = yφ))
2 sb56 1762 . 2 (x(x = y φ) ↔ x(x = yφ))
31, 2bitr4i 176 1 ([y / x]φx(x = y φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378  [wsb 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-sb 1643 This theorem is referenced by:  sbnv  1765  sborv  1767  sbi2v  1769  nfsbxy  1815  nfsbxyt  1816  2sb5  1856  dfsb7  1864  sb7f  1865  sbexyz  1876  sbc5  2781
 Copyright terms: Public domain W3C validator