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Theorem equsb3 2105
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 2033 . 2 (𝑥 = 𝑤 → (𝑥 = 𝑧𝑤 = 𝑧))
2 equequ1 2033 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
31, 2sbievw2 2103 1 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
Colors of variables: wff setvar class
Syntax hints:  wb 209  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071
This theorem is referenced by:  equsb1v  2107  mo3  2563  sb8eulem  2597  sb8iota  6350  mo5f  30556  mptsnunlem  35246  wl-equsb3  35448  wl-mo3t  35468  wl-sb8eut  35469  frege55lem1b  41180  sbeqal1  41689  icheq  44587
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