ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqsb3 GIF version

Theorem eqsb3 2157
Description: Substitution applied to an atomic wff (class version of equsb3 1841). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eqsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqsb3lem 2156 . . 3 ([𝑤 / 𝑦]𝑦 = 𝐴𝑤 = 𝐴)
21sbbii 1664 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴)
3 nfv 1437 . . 3 𝑤 𝑦 = 𝐴
43sbco2 1855 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴)
5 eqsb3lem 2156 . 2 ([𝑥 / 𝑤]𝑤 = 𝐴𝑥 = 𝐴)
62, 4, 53bitr3i 203 1 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-cleq 2049
This theorem is referenced by:  pm13.183  2704  eqsbc3  2825
  Copyright terms: Public domain W3C validator