Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-eleq1w Structured version   Visualization version   GIF version

Theorem bj-eleq1w 31871
Description: Weaker version of eleq1 2580 (but more general than elequ1 1945) not depending on ax-ext 2494 (nor ax-12 1983 nor df-cleq 2507). Remark: this can also be done with eleq1i 2583, eqeltri 2588, eqeltrri 2589, eleq1a 2587, eleq1d 2576, eqeltrd 2592, eqeltrrd 2593, eqneltrd 2611, eqneltrrd 2612, nelneq 2616. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-eleq1w (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))

Proof of Theorem bj-eleq1w
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ2 1903 . . . 4 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
21anbi1d 736 . . 3 (𝑥 = 𝑦 → ((𝑧 = 𝑥𝑧𝐴) ↔ (𝑧 = 𝑦𝑧𝐴)))
32exbidv 1803 . 2 (𝑥 = 𝑦 → (∃𝑧(𝑧 = 𝑥𝑧𝐴) ↔ ∃𝑧(𝑧 = 𝑦𝑧𝐴)))
4 df-clel 2510 . 2 (𝑥𝐴 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝐴))
5 df-clel 2510 . 2 (𝑦𝐴 ↔ ∃𝑧(𝑧 = 𝑦𝑧𝐴))
63, 4, 53bitr4g 301 1 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wex 1694  wcel 1938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-clel 2510
This theorem is referenced by:  bj-clelsb3  31873  bj-nfcjust  31875
  Copyright terms: Public domain W3C validator