Theorem sbal2 2449

Description: Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 3-Oct-2018.)

Assertion

Ref	Expression
sbal2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Distinct variable group:   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbal2

Step	Hyp	Ref	Expression
1		sb4b 2346	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
2	1	adantl 481	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
3		nfnae 2306	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
4		sb4b 2346	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜑)))
5	3, 4	albid 2077	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑)))
6		alcom 2024	. . . . . 6 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑))
7		nfnae 2306	. . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
8		nfeqf1 2287	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
9		19.21t 2061	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑦 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
10	8, 9	syl 17	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑦 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
11	7, 10	albid 2077	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
12	6, 11	syl5bb 271	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
13	5, 12	sylan9bbr 733	. . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
14	2, 13	bitr4d 270	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
15	14	ex 449	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)))
16		sbid 2100	. . . 4 ⊢ ([𝑦 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥𝜑)
17		drsb2 2366	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → ([𝑦 / 𝑦]∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
18	16, 17	syl5bbr 273	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
19		sbid 2100	. . . . 5 ⊢ ([𝑦 / 𝑦]𝜑 ↔ 𝜑)
20		drsb2 2366	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → ([𝑦 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑))
21	19, 20	syl5bbr 273	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
22	21	dral2 2312	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
23	18, 22	bitr3d 269	. 2 ⊢ (∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
24	15, 23	pm2.61d2 171	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  Ⅎwnf 1699  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by:  2sb5ndVD  38168  2sb5ndALT  38190