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Theorem sbcom2 2168
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.
StepHypRef Expression
1 2sb6 2094 . . . . . . . . 9 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑧𝑥((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑))
2 alcom 2163 . . . . . . . . 9 (∀𝑧𝑥((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥𝑧((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑))
3 ancomst 467 . . . . . . . . . 10 (((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
432albii 1821 . . . . . . . . 9 (∀𝑥𝑧((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
51, 2, 43bitri 299 . . . . . . . 8 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
6 2sb6 2094 . . . . . . . 8 ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
75, 6bitr4i 280 . . . . . . 7 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑧]𝜑)
8 sbequ 2090 . . . . . . . 8 (𝑢 = 𝑦 → ([𝑢 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
98sbbidv 2084 . . . . . . 7 (𝑢 = 𝑦 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
107, 9syl5bbr 287 . . . . . 6 (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
11 sbequ 2090 . . . . . 6 (𝑣 = 𝑤 → ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
1210, 11sylan9bb 512 . . . . 5 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
13 sbequ 2090 . . . . . . 7 (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
1413sbbidv 2084 . . . . . 6 (𝑣 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
15 sbequ 2090 . . . . . 6 (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
1614, 15sylan9bbr 513 . . . . 5 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
1712, 16bitr3d 283 . . . 4 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
1817ex 415 . . 3 (𝑢 = 𝑦 → (𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
19 ax6ev 1972 . . 3 𝑢 𝑢 = 𝑦
2018, 19exlimiiv 1932 . 2 (𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
21 ax6ev 1972 . 2 𝑣 𝑣 = 𝑤
2220, 21exlimiiv 1932 1 ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1535  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2161
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070
This theorem is referenced by:  sbco4lem  2283  sbco4  2284  2mo  2732  cnvopab  5973  2reu8i  43460  ichcom  43767  ichbi12i  43768
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