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Theorem sb6 1763
 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 1762 . . 3 (x(x = y φ) ↔ x(x = yφ))
21anbi2i 430 . 2 (((x = yφ) x(x = y φ)) ↔ ((x = yφ) x(x = yφ)))
3 df-sb 1643 . 2 ([y / x]φ ↔ ((x = yφ) x(x = y φ)))
4 ax-4 1397 . . 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 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:  sb5  1764  sbnv  1765  sbanv  1766  sbi1v  1768  sbi2v  1769  hbs1  1811  2sb6  1857  sbcom2v  1858  sb6a  1861  sb7af  1866  sbalyz  1872  sbal1yz  1874  exsb  1881  sbal2  1895
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