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Theorem sb56 1762
 Description: Two equivalent ways of expressing the proper substitution of y for x in φ, when x and y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1643. (Contributed by NM, 14-Apr-2008.)
Assertion
Ref Expression
sb56 (x(x = y φ) ↔ x(x = yφ))
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem sb56
StepHypRef Expression
1 hba1 1430 . 2 (x(x = yφ) → xx(x = yφ))
2 ax11v 1705 . . 3 (x = y → (φx(x = yφ)))
3 ax-4 1397 . . . 4 (x(x = yφ) → (x = yφ))
43com12 27 . . 3 (x = y → (x(x = yφ) → φ))
52, 4impbid 120 . 2 (x = y → (φx(x = yφ)))
61, 5equsex 1613 1 (x(x = y φ) ↔ x(x = yφ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378 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 This theorem is referenced by:  sb6  1763  sb5  1764  alexeq  2664
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