Theorem dfsb2 2373

Description: An alternate definition of proper substitution that, like df-sb 1881, 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 = 𝑦 → 𝜑)))

Proof of Theorem dfsb2

Step	Hyp	Ref	Expression
1		sp 2053	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2		sbequ2 1882	. . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
3	2	sps 2055	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
4		orc 400	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	1, 3, 4	syl6an 568	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))))
6		sb4 2356	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7		olc 399	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	6, 7	syl6 35	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))))
9	5, 8	pm2.61i 176	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
10		sbequ1 2110	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
11	10	imp 445	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)
12		sb2 2352	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
13	11, 12	jaoi 394	. 2 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)) → [𝑦 / 𝑥]𝜑)
14	9, 13	impbii 199	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384  ∀wal 1481  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  dfsb3  2374