Theorem sb4b 2509

Description: Simplified definition of substitution when variables are distinct. Version of sb6 2122 with a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by NM, 27-May-1997.) Revise df-sb 2100. (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 2190	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑡
2		nfeqf2 2411	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
3	1, 2	nfan1 2239	. . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡)
4		equequ2 2059	. . . . . . 7 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
5	4	imbi1d 344	. . . . . 6 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
6	5	adantl 487	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
7	3, 6	albid 2261	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
8	7	pm5.74da 816	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
9	8	albidv 1953	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
10		dfsb 2101	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
11		ax6ev 2002	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑡
12	11	a1bi 365	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
13		19.23v 1975	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
14	12, 13	bitr4i 281	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
15	9, 10, 14	3bitr4g 317	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  sb3b  2510  sb2  2513  sb4a  2514  hbsb2  2516  dfsb2  2527  sbcom3  2540  sbal1  2562  sbal2  2563  wl-2sb6d  38272  wl-sbalnae  38276