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Theorem equsb3 2106
 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 2032 . 2 (𝑥 = 𝑤 → (𝑥 = 𝑧𝑤 = 𝑧))
2 equequ1 2032 . 2 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
31, 2sbievw2 2104 1 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070 This theorem is referenced by:  equsb3rOLD  2108  equsb1v  2109  mo3  2623  sb8eulem  2659  sb8iota  6299  mo5f  30301  mptsnunlem  34822  wl-equsb3  35024  wl-mo3t  35044  wl-sb8eut  35045  wl-dfrmosb  35085  frege55lem1b  40683  sbeqal1  41189  icheq  44063
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