Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfss1OLD Structured version   Visualization version   GIF version

Theorem dfss1OLD 3780
 Description: Obsolete as of 22-Jul-2021. (Contributed by NM, 10-Jan-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
dfss1OLD (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Proof of Theorem dfss1OLD
StepHypRef Expression
1 df-ss 3554 . 2 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
2 incom 3767 . . 3 (𝐴𝐵) = (𝐵𝐴)
32eqeq1i 2615 . 2 ((𝐴𝐵) = 𝐴 ↔ (𝐵𝐴) = 𝐴)
41, 3bitri 263 1 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∩ cin 3539   ⊆ wss 3540 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator