MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqsbc1 Structured version   Visualization version   GIF version

Theorem eqsbc1 3834
Description: Substitution for the left-hand side in an equality. Class version of eqsb1 2866. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2376. (Revised by Wolf Lammen, 29-Apr-2023.)
Assertion
Ref Expression
eqsbc1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3789 . 2 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝑥 = 𝐵[𝐴 / 𝑥]𝑥 = 𝐵))
2 eqeq1 2740 . 2 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐴 = 𝐵))
3 sbsbc 3791 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵[𝑦 / 𝑥]𝑥 = 𝐵)
4 eqsb1 2866 . . 3 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
53, 4bitr3i 277 . 2 ([𝑦 / 𝑥]𝑥 = 𝐵𝑦 = 𝐵)
61, 2, 5vtoclbg 3556 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  [wsb 2063  wcel 2107  [wsbc 3787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-sbc 3788
This theorem is referenced by:  sbceqalOLD  3851  eqsbc2  3853  fmptsnd  7190  fvmptnn04if  22856  snfil  23873  f1omptsnlem  37338  mptsnunlem  37340  topdifinffinlem  37349  relowlpssretop  37366  iotavalb  44454  onfrALTlem5  44567  eqsbc2VD  44865  onfrALTlem5VD  44910
  Copyright terms: Public domain W3C validator