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Theorem dfsb3 2378
Description: An alternate definition of proper substitution df-sb 1883 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)
Assertion
Ref Expression
dfsb3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem dfsb3
StepHypRef Expression
1 df-or 385 . 2 (((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (¬ (𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
2 dfsb2 2377 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑)))
3 imnan 438 . . 3 ((𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ (𝑥 = 𝑦𝜑))
43imbi1i 339 . 2 (((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (¬ (𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
51, 2, 43bitr4i 292 1 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1478  [wsb 1882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1883
This theorem is referenced by:  sbn  2395
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