Theorem dfsb2 1223

Description: An alternate definition of proper substitution that, like df-sb 1170, mixes free and bound variables to avoid distinct variable requirements.

Assertion

Ref	Expression
dfsb2	⊢ ([y / x]φ ↔ ((x = y ⋀ φ) ⋁ ∀x(x = y → φ)))

Proof of Theorem dfsb2

Step	Hyp	Ref	Expression
1		sbequ2 1177	. . . . . 6 ⊢ (x = y → ([y / x]φ → φ))
2	1	a4s 982	. . . . 5 ⊢ (∀x x = y → ([y / x]φ → φ))
3		ax-4 971	. . . . 5 ⊢ (∀x x = y → x = y)
4	2, 3	jctild 600	. . . 4 ⊢ (∀x x = y → ([y / x]φ → (x = y ⋀ φ)))
5		orc 269	. . . 4 ⊢ ((x = y ⋀ φ) → ((x = y ⋀ φ) ⋁ ∀x(x = y → φ)))
6	4, 5	syl6 22	. . 3 ⊢ (∀x x = y → ([y / x]φ → ((x = y ⋀ φ) ⋁ ∀x(x = y → φ))))
7		sb4 1221	. . . 4 ⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))
8		olc 268	. . . 4 ⊢ (∀x(x = y → φ) → ((x = y ⋀ φ) ⋁ ∀x(x = y → φ)))
9	7, 8	syl6 22	. . 3 ⊢ (¬ ∀x x = y → ([y / x]φ → ((x = y ⋀ φ) ⋁ ∀x(x = y → φ))))
10	6, 9	pm2.61i 126	. 2 ⊢ ([y / x]φ → ((x = y ⋀ φ) ⋁ ∀x(x = y → φ)))
11		sbequ1 1176	. . . 4 ⊢ (x = y → (φ → [y / x]φ))
12	11	imp 350	. . 3 ⊢ ((x = y ⋀ φ) → [y / x]φ)
13		sb2 1175	. . 3 ⊢ (∀x(x = y → φ) → [y / x]φ)
14	12, 13	jaoi 341	. 2 ⊢ (((x = y ⋀ φ) ⋁ ∀x(x = y → φ)) → [y / x]φ)
15	10, 14	impbi 157	1 ⊢ ([y / x]φ ↔ ((x = y ⋀ φ) ⋁ ∀x(x = y → φ)))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223  ∀wal 952   = wceq 954  [wsbc 1168

This theorem is referenced by:  dfsb3 1224

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-10 964  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170

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