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Theorem sb56 2098
 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 1649. (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 nfa1 1788 . 2 xx(x = yφ)
2 ax11v 2096 . . 3 (x = y → (φx(x = yφ)))
3 sp 1747 . . . 4 (x(x = yφ) → (x = yφ))
43com12 27 . . 3 (x = y → (x(x = yφ) → φ))
52, 4impbid 183 . 2 (x = y → (φx(x = yφ)))
61, 5equsex 1962 1 (x(x = y φ) ↔ x(x = yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  sb6  2099  sb5  2100  alexeq  2968
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