Theorem sbal2 2561

Description: Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 24-Dec-2022.) (Proof shortened by Wolf Lammen, 23-Sep-2023.) Usage of this theorem is discouraged because it depends on ax-13 2404. Use sbal 2204 instead. (New usage is discouraged.)

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		sbequ12 2287	. . . . 5 ⊢ (𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
2	1	sps 2221	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ [𝑧 / 𝑦]∀𝑥𝜑))
3		sbequ12 2287	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
4	3	sps 2221	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
5	4	dral2 2470	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑧 → (∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
6	2, 5	bitr3d 284	. . 3 ⊢ (∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
7	6	adantl 486	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
8		sb4b 2507	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
9	8	adantl 486	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
10		nfnae 2466	. . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑦 𝑦 = 𝑧
11		sb4b 2507	. . . . . 6 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ([𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → 𝜑)))
12	10, 11	albid 2258	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑)))
13		alcom 2194	. . . . 5 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑))
14	12, 13	bitrdi 290	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑧 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑)))
15		nfnae 2466	. . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
16		nfeqf1 2411	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧)
17		19.21t 2242	. . . . . 6 ⊢ (Ⅎ𝑥 𝑦 = 𝑧 → (∀𝑥(𝑦 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
18	16, 17	syl 18	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑦 = 𝑧 → 𝜑) ↔ (𝑦 = 𝑧 → ∀𝑥𝜑)))
19	15, 18	albid 2258	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥(𝑦 = 𝑧 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
20	14, 19	sylan9bbr 519	. . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑧 → ∀𝑥𝜑)))
21	9, 20	bitr4d 285	. 2 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
22	7, 21	pm2.61dan 824	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  Ⅎwnf 1813  [wsb 2096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097

This theorem is referenced by:  2sb5ndVD  45601  2sb5ndALT  45623