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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.
StepHypRef Expression
1 ax6ev 2079 . 2 𝑢 𝑢 = 𝑦
2 ax6ev 2079 . 2 𝑣 𝑣 = 𝑤
3 2sb6 2312 . . . . . . . . . 10 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑧𝑥((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑))
4 alcom 2211 . . . . . . . . . 10 (∀𝑧𝑥((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥𝑧((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑))
5 ancomst 458 . . . . . . . . . . 11 (((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
652albii 1921 . . . . . . . . . 10 (∀𝑥𝑧((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
73, 4, 63bitri 289 . . . . . . . . 9 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
8 2sb6 2312 . . . . . . . . 9 ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
97, 8bitr4i 270 . . . . . . . 8 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑧]𝜑)
10 nfv 2015 . . . . . . . . 9 𝑧 𝑢 = 𝑦
11 sbequ 2507 . . . . . . . . 9 (𝑢 = 𝑦 → ([𝑢 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
1210, 11sbbid 2286 . . . . . . . 8 (𝑢 = 𝑦 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
139, 12syl5bbr 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))
1513, 14sylan9bb 507 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
16 nfv 2015 . . . . . . . 8 𝑥 𝑣 = 𝑤
17 sbequ 2507 . . . . . . . 8 (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
1816, 17sbbid 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))
2018, 19sylan9bbr 508 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
2115, 20bitr3d 273 . . . . 5 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
2221ex 403 . . . 4 (𝑢 = 𝑦 → (𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
2322exlimdv 2034 . . 3 (𝑢 = 𝑦 → (∃𝑣 𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
2423exlimiv 2031 . 2 (∃𝑢 𝑢 = 𝑦 → (∃𝑣 𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
251, 2, 24mp2 9 1 ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)
 Colors of variables: wff setvar class 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)
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