Theorem sbcom2OLD 2579

Description: Obsolete version of sbcom2 2578 as of 23-Dec-2022. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 24-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcom2OLD	⊢ ([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 sbcom2OLD

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

Step	Hyp	Ref	Expression
1		ax6ev 2079	. 2 ⊢ ∃𝑢 𝑢 = 𝑦
2		ax6ev 2079	. 2 ⊢ ∃𝑣 𝑣 = 𝑤
3		2sb6 2312	. . . . . . . . . 10 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑))
4		alcom 2211	. . . . . . . . . 10 ⊢ (∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥∀𝑧((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑))
5		ancomst 458	. . . . . . . . . . 11 ⊢ (((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
6	5	2albii 1921	. . . . . . . . . 10 ⊢ (∀𝑥∀𝑧((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
7	3, 4, 6	3bitri 289	. . . . . . . . 9 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
8		2sb6 2312	. . . . . . . . 9 ⊢ ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
9	7, 8	bitr4i 270	. . . . . . . 8 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑧]𝜑)
10		nfv 2015	. . . . . . . . 9 ⊢ Ⅎ𝑧 𝑢 = 𝑦
11		sbequ 2507	. . . . . . . . 9 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
12	10, 11	sbbid 2286	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
13	9, 12	syl5bbr 277	. . . . . . 7 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
14		sbequ 2507	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
15	13, 14	sylan9bb 507	. . . . . 6 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
16		nfv 2015	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑣 = 𝑤
17		sbequ 2507	. . . . . . . 8 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
18	16, 17	sbbid 2286	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
19		sbequ 2507	. . . . . . 7 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
20	18, 19	sylan9bbr 508	. . . . . 6 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
21	15, 20	bitr3d 273	. . . . 5 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
22	21	ex 403	. . . 4 ⊢ (𝑢 = 𝑦 → (𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
23	22	exlimdv 2034	. . 3 ⊢ (𝑢 = 𝑦 → (∃𝑣 𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
24	23	exlimiv 2031	. 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 198   ∧ wa 386  ∀wal 1656  ∃wex 1880  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-nf 1885  df-sb 2070

This theorem is referenced by: (None)