Theorem sb4b 2473

Description: Simplified definition of substitution when variables are distinct. Version of sb6 2087 with a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 27-May-1997.) Revise df-sb 2067. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.)

Assertion

Ref	Expression
sb4b	⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Proof of Theorem sb4b

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfna1 2148	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑡
2		nfeqf2 2375	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
3	1, 2	nfan1 2192	. . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡)
4		equequ2 2028	. . . . . . 7 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
5	4	imbi1d 341	. . . . . 6 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
6	5	adantl 482	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
7	3, 6	albid 2214	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
8	7	pm5.74da 801	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
9	8	albidv 1922	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
10		df-sb 2067	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
11		ax6ev 1972	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑡
12	11	a1bi 362	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
13		19.23v 1944	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
14	12, 13	bitr4i 277	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
15	9, 10, 14	3bitr4g 313	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1538  ∃wex 1780  [wsb 2066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-12 2170  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1781  df-nf 1785  df-sb 2067

This theorem is referenced by:  sb3b  2475  sb2  2478  sb1OLD  2480  sb4a  2482  hbsb2  2484  dfsb2  2495  sbcom3  2508  sbal1  2531  sbal2  2532  wl-2sb6d  35818  wl-sbalnae  35822