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Theorem dfsb2 2055
 Description: An alternate definition of proper substitution that, like df-sb 1649, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)
Assertion
Ref Expression
dfsb2 ([y / x]φ ↔ ((x = y φ) x(x = yφ)))

Proof of Theorem dfsb2
StepHypRef Expression
1 sp 1747 . . . 4 (x x = yx = y)
2 sbequ2 1650 . . . . 5 (x = y → ([y / x]φφ))
32sps 1754 . . . 4 (x x = y → ([y / x]φφ))
4 orc 374 . . . 4 ((x = y φ) → ((x = y φ) x(x = yφ)))
51, 3, 4ee12an 1363 . . 3 (x x = y → ([y / x]φ → ((x = y φ) x(x = yφ))))
6 sb4 2053 . . . 4 x x = y → ([y / x]φx(x = yφ)))
7 olc 373 . . . 4 (x(x = yφ) → ((x = y φ) x(x = yφ)))
86, 7syl6 29 . . 3 x x = y → ([y / x]φ → ((x = y φ) x(x = yφ))))
95, 8pm2.61i 156 . 2 ([y / x]φ → ((x = y φ) x(x = yφ)))
10 sbequ1 1918 . . . 4 (x = y → (φ → [y / x]φ))
1110imp 418 . . 3 ((x = y φ) → [y / x]φ)
12 sb2 2023 . . 3 (x(x = yφ) → [y / x]φ)
1311, 12jaoi 368 . 2 (((x = y φ) x(x = yφ)) → [y / x]φ)
149, 13impbii 180 1 ([y / x]φ ↔ ((x = y φ) x(x = yφ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  [wsb 1648 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  dfsb3  2056
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