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Theorem dfsb2 2498
Description: An alternate definition of proper substitution that, like dfsb1 2486, mixes free and bound variables to avoid distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 2373. (Contributed by NM, 17-Feb-2005.) (New usage is discouraged.)
Assertion
Ref Expression
dfsb2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem dfsb2
StepHypRef Expression
1 sp 2179 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 sbequ2 2244 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
32sps 2181 . . . 4 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
4 orc 863 . . . 4 ((𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
51, 3, 4syl6an 680 . . 3 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑))))
6 sb4b 2476 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
7 olc 864 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
86, 7syl6bi 252 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑))))
95, 8pm2.61i 182 . 2 ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
10 sbequ1 2243 . . . 4 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
1110imp 406 . . 3 ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
12 sb2 2481 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
1311, 12jaoi 853 . 2 (((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)) → [𝑦 / 𝑥]𝜑)
149, 13impbii 208 1 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 843  wal 1539  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174  ax-13 2373
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790  df-sb 2071
This theorem is referenced by:  dfsb3  2499
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