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 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Proof of Theorem dfsb2

Step	Hyp	Ref	Expression
1		sp 2179	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦)
2		sbequ2 2244	. . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
3	2	sps 2181	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
4		orc 863	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	1, 3, 4	syl6an 680	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))))
6		sb4b 2476	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7		olc 864	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	6, 7	syl6bi 252	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))))
9	5, 8	pm2.61i 182	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
10		sbequ1 2243	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
11	10	imp 406	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)
12		sb2 2481	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
13	11, 12	jaoi 853	. 2 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)) → [𝑦 / 𝑥]𝜑)
14	9, 13	impbii 208	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))

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