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Theorem dfsb2 2360
Description: An alternate definition of proper substitution that, like df-sb 1867, mixes free and bound variables to avoid distinct variable requirements. (Contributed by NM, 17-Feb-2005.)
Assertion
Ref Expression
dfsb2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem dfsb2
StepHypRef Expression
1 sp 2039 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 sbequ2 1868 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
32sps 2041 . . . 4 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
4 orc 398 . . . 4 ((𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
51, 3, 4syl6an 565 . . 3 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑))))
6 sb4 2343 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
7 olc 397 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
86, 7syl6 34 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑))))
95, 8pm2.61i 174 . 2 ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
10 sbequ1 2094 . . . 4 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
1110imp 443 . . 3 ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
12 sb2 2339 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
1311, 12jaoi 392 . 2 (((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)) → [𝑦 / 𝑥]𝜑)
149, 13impbii 197 1 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  wal 1472  [wsb 1866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-12 2032  ax-13 2232
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867
This theorem is referenced by:  dfsb3  2361
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