Theorem dfsb3 2525

Description: An alternate definition of proper substitution df-sb 2091 that uses only primitive connectives (no defined terms) on the right-hand side. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by NM, 6-Mar-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
dfsb3	⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Proof of Theorem dfsb3

Step	Hyp	Ref	Expression
1		df-or 859	. 2 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (¬ (𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		dfsb2 2524	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		imnan 403	. . 3 ⊢ ((𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ (𝑥 = 𝑦 ∧ 𝜑))
4	3	imbi1i 351	. 2 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (¬ (𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	1, 2, 4	3bitr4i 305	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858  ∀wal 1558  [wsb 2090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-12 2212  ax-13 2403

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1800  df-nf 1804  df-sb 2091

This theorem is referenced by: (None)