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Theorem eqsb3 2936
Description: Substitution applied to an atomic wff (class version of equsb3 2100). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem eqsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2822 . 2 (𝑥 = 𝑤 → (𝑥 = 𝐴𝑤 = 𝐴))
2 eqeq1 2822 . 2 (𝑤 = 𝑦 → (𝑤 = 𝐴𝑦 = 𝐴))
31, 2sbievw2 2098 1 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-cleq 2811
This theorem is referenced by:  pm13.183  3656  pm13.183OLD  3657  eqsbc3  3814  eqsbc3OLD  3815
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