Theorem sbcom2 2164

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, 23-Dec-2022.)

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		2sb6 2090	. . . . . . . . 9 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑))
2		alcom 2159	. . . . . . . . 9 ⊢ (∀𝑧∀𝑥((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥∀𝑧((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑))
3		ancomst 467	. . . . . . . . . 10 ⊢ (((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
4	3	2albii 1817	. . . . . . . . 9 ⊢ (∀𝑥∀𝑧((𝑧 = 𝑣 ∧ 𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
5	1, 2, 4	3bitri 299	. . . . . . . 8 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
6		2sb6 2090	. . . . . . . 8 ⊢ ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ ∀𝑥∀𝑧((𝑥 = 𝑢 ∧ 𝑧 = 𝑣) → 𝜑))
7	5, 6	bitr4i 280	. . . . . . 7 ⊢ ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑧]𝜑)
8		sbequ 2086	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
9	8	sbbidv 2080	. . . . . . 7 ⊢ (𝑢 = 𝑦 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
10	7, 9	syl5bbr 287	. . . . . 6 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
11		sbequ 2086	. . . . . 6 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
12	10, 11	sylan9bb 512	. . . . 5 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
13		sbequ 2086	. . . . . . 7 ⊢ (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
14	13	sbbidv 2080	. . . . . 6 ⊢ (𝑣 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
15		sbequ 2086	. . . . . 6 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
16	14, 15	sylan9bbr 513	. . . . 5 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
17	12, 16	bitr3d 283	. . . 4 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑤) → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
18	17	ex 415	. . 3 ⊢ (𝑢 = 𝑦 → (𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
19		ax6ev 1968	. . 3 ⊢ ∃𝑢 𝑢 = 𝑦
20	18, 19	exlimiiv 1928	. 2 ⊢ (𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
21		ax6ev 1968	. 2 ⊢ ∃𝑣 𝑣 = 𝑤
22	20, 21	exlimiiv 1928	1 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-11 2157

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066

This theorem is referenced by:  sbco4lem  2279  sbco4  2280  2mo  2729  cnvopab  5996  2reu8i  43311  dfich2bi  43614  ichcom  43616  ichbi12i  43617