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Theorem equsb3 2431
Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 2031. (Revised by Wolf Lammen, 21-Sep-2018.)
Assertion
Ref Expression
equsb3 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Distinct variable group:   𝑦,𝑧

Proof of Theorem equsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equsb3lem 2430 . . 3 ([𝑤 / 𝑦]𝑦 = 𝑧𝑤 = 𝑧)
21sbbii 1884 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3 sbcom3 2410 . . 3 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧)
4 nfv 1840 . . . 4 𝑤[𝑥 / 𝑦]𝑦 = 𝑧
54sbf 2379 . . 3 ([𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
63, 5bitri 264 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
7 equsb3lem 2430 . 2 ([𝑥 / 𝑤]𝑤 = 𝑧𝑥 = 𝑧)
82, 6, 73bitr3i 290 1 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  sb8eu  2502  mo3  2506  sb8iota  5827  mo5f  29213  mptsnunlem  32856  wl-equsb3  33008  wl-mo3t  33029  wl-sb8eut  33030  frege55lem1b  37710  sbeqal1  38119
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