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Theorem sbcom2 2574
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.
StepHypRef Expression
1 ax6ev 2048 . 2 𝑢 𝑢 = 𝑦
2 ax6ev 2048 . 2 𝑣 𝑣 = 𝑤
3 2sb6 2573 . . . . . . . . . 10 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑧𝑥((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑))
4 alcom 2178 . . . . . . . . . 10 (∀𝑧𝑥((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥𝑧((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑))
5 ancomst 467 . . . . . . . . . . 11 (((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
652albii 1889 . . . . . . . . . 10 (∀𝑥𝑧((𝑧 = 𝑣𝑥 = 𝑢) → 𝜑) ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
73, 4, 63bitri 286 . . . . . . . . 9 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
8 2sb6 2573 . . . . . . . . 9 ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ ∀𝑥𝑧((𝑥 = 𝑢𝑧 = 𝑣) → 𝜑))
97, 8bitr4i 267 . . . . . . . 8 ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑢 / 𝑥][𝑣 / 𝑧]𝜑)
10 nfv 1984 . . . . . . . . 9 𝑧 𝑢 = 𝑦
11 sbequ 2505 . . . . . . . . 9 (𝑢 = 𝑦 → ([𝑢 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
1210, 11sbbid 2532 . . . . . . . 8 (𝑢 = 𝑦 → ([𝑣 / 𝑧][𝑢 / 𝑥]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
139, 12syl5bbr 274 . . . . . . 7 (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑣 / 𝑧][𝑦 / 𝑥]𝜑))
14 sbequ 2505 . . . . . . 7 (𝑣 = 𝑤 → ([𝑣 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
1513, 14sylan9bb 738 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
16 nfv 1984 . . . . . . . 8 𝑥 𝑣 = 𝑤
17 sbequ 2505 . . . . . . . 8 (𝑣 = 𝑤 → ([𝑣 / 𝑧]𝜑 ↔ [𝑤 / 𝑧]𝜑))
1816, 17sbbid 2532 . . . . . . 7 (𝑣 = 𝑤 → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑢 / 𝑥][𝑤 / 𝑧]𝜑))
19 sbequ 2505 . . . . . . 7 (𝑢 = 𝑦 → ([𝑢 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
2018, 19sylan9bbr 739 . . . . . 6 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑢 / 𝑥][𝑣 / 𝑧]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
2115, 20bitr3d 270 . . . . 5 ((𝑢 = 𝑦𝑣 = 𝑤) → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
2221ex 449 . . . 4 (𝑢 = 𝑦 → (𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
2322exlimdv 2002 . . 3 (𝑢 = 𝑦 → (∃𝑣 𝑣 = 𝑤 → ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)))
2423exlimiv 1999 . 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 196  wa 383  wal 1622  wex 1845  [wsb 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1846  df-nf 1851  df-sb 2039
This theorem is referenced by:  sbco4lem  2594  sbco4  2595  2mo  2681  cnvopab  5683
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