Theorem sbcom2 2433

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Sep-2018.)

Assertion

Ref	Expression
sbcom2	⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Distinct variable groups:   𝑥,𝑧   𝑥,𝑤   𝑦,𝑧

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

Proof of Theorem sbcom2

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6ev 1877	. 2 ⊢ ∃𝑢 𝑢 = 𝑦
2		ax6ev 1877	. 2 ⊢ ∃𝑣 𝑣 = 𝑤
3		2sb6 2432	. . . . . . . . . 10 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑))
4		alcom 2024	. . . . . . . . . 10 ⊢ (∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥∀𝑧((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑))
5		ancomst 467	. . . . . . . . . . 11 ⊢ (((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
6	5	2albii 1738	. . . . . . . . . 10 ⊢ (∀𝑥∀𝑧((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
7	3, 4, 6	3bitri 285	. . . . . . . . 9 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
8		2sb6 2432	. . . . . . . . 9 ⊢ ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
9	7, 8	bitr4i 266	. . . . . . . 8 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑧]𝜑)
10		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑧 𝑢 = 𝑦
11		sbequ 2364	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
12	10, 11	sbbid 2391	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
13	9, 12	syl5bbr 273	. . . . . . 7 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
14		sbequ 2364	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
15	13, 14	sylan9bb 732	. . . . . 6 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
16		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣 = 𝑤
17		sbequ 2364	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
18	16, 17	sbbid 2391	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
19		sbequ 2364	. . . . . . 7 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
20	18, 19	sylan9bbr 733	. . . . . 6 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
21	15, 20	bitr3d 269	. . . . 5 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
22	21	ex 449	. . . 4 ⊢ (𝑢 = 𝑦 → (𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
23	22	exlimdv 1848	. . 3 ⊢ (𝑢 = 𝑦 → (∃𝑣 𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
24	23	exlimiv 1845	. 2 ⊢ (∃𝑢 𝑢 = 𝑦 → (∃𝑣 𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
25	1, 2, 24	mp2 9	1 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  [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:  sbco4lem  2453  sbco4  2454  2mo  2539  cnvopab  5452