MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsb2 Structured version   Visualization version   GIF version

Theorem dfsb2 2449
Description: An alternate definition of proper substitution that, like df-sb 2012, 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 2167 . . . 4 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
2 sbequ2 2013 . . . . 5 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
32sps 2169 . . . 4 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
4 orc 856 . . . 4 ((𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
51, 3, 4syl6an 674 . . 3 (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑))))
6 sb4 2432 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
7 olc 857 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
86, 7syl6 35 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑))))
95, 8pm2.61i 177 . 2 ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
10 sbequ1 2228 . . . 4 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
1110imp 397 . . 3 ((𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
12 sb2 2427 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
1311, 12jaoi 846 . 2 (((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)) → [𝑦 / 𝑥]𝜑)
149, 13impbii 201 1 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 836  wal 1599  [wsb 2011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163  ax-13 2334
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012
This theorem is referenced by:  dfsb3  2450
  Copyright terms: Public domain W3C validator