 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqsb3lem GIF version

Theorem eqsb3lem 2140
 Description: Lemma for eqsb3 2141. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
eqsb3lem ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eqsb3lem
StepHypRef Expression
1 nfv 1421 . 2 𝑦 𝑥 = 𝐴
2 eqeq1 2046 . 2 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
31, 2sbie 1674 1 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033 This theorem is referenced by:  eqsb3  2141
 Copyright terms: Public domain W3C validator