 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqsb3 Structured version   Visualization version   GIF version

Theorem eqsb3 2715
 Description: Substitution applied to an atomic wff (class version of equsb3 2420). (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 2714 . . 3 ([𝑤 / 𝑦]𝑦 = 𝐴𝑤 = 𝐴)
21sbbii 1874 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴)
3 nfv 1830 . . 3 𝑤 𝑦 = 𝐴
43sbco2 2403 . 2 ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴)
5 eqsb3lem 2714 . 2 ([𝑥 / 𝑤]𝑤 = 𝐴𝑥 = 𝐴)
62, 4, 53bitr3i 289 1 ([𝑥 / 𝑦]𝑦 = 𝐴𝑥 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603 This theorem is referenced by:  pm13.183  3313  eqsbc3  3442
 Copyright terms: Public domain W3C validator