ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb6 Structured version   GIF version

Theorem sb6 1744
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.) (Revised by NM, 14-Apr-2008.)
Assertion
Ref Expression
sb6 ([y / x]φx(x = yφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem sb6
StepHypRef Expression
1 sb56 1743 . . 3 (x(x = y φ) ↔ x(x = yφ))
21anbi2i 433 . 2 (((x = yφ) x(x = y φ)) ↔ ((x = yφ) x(x = yφ)))
3 df-sb 1624 . 2 ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
4 ax-4 1377 . . 3 (x(x = yφ) → (x = yφ))
54pm4.71ri 372 . 2 (x(x = yφ) ↔ ((x = yφ) x(x = yφ)))
62, 3, 53bitr4i 201 1 ([y / x]φx(x = yφ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1224  wex 1358  [wsb 1623
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 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405
This theorem depends on definitions:  df-bi 110  df-sb 1624
This theorem is referenced by:  sb5  1745  sbnv  1746  sbanv  1747  sbi1v  1749  sbi2v  1750  hbs1  1792  2sb6  1838  sbcom2v  1839  sb6a  1842  sb7af  1847  sbalyz  1853  sbal1yz  1855  exsb  1862  sbal2  1876
  Copyright terms: Public domain W3C validator