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Theorem equsb3 2102
Description: Substitution in an equality. (Contributed by Raph Levien and FL, 4-Dec-2005.) Reduce axiom usage. (Revised by Wolf Lammen, 23-Jul-2023.)
Assertion
Ref Expression
equsb3 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
Distinct variable group:   𝑥,𝑧

Proof of Theorem equsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equequ1 2029 . 2 (𝑥 = 𝑤 → (𝑥 = 𝑧𝑤 = 𝑧))
2 equequ1 2029 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
31, 2sbievw2 2100 1 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069
This theorem is referenced by:  equsb1v  2104  mo3  2559  sb8eulem  2593  sb8iota  6508  mo5f  31729  mptsnunlem  36219  wl-equsb3  36421  wl-mo3t  36441  wl-sb8eut  36442  frege55lem1b  42646  sbeqal1  43157  icheq  46130
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