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

Theorem bj-rcleq 34744
 Description: Relative version of dfcleq 2752. (Contributed by BJ, 27-Dec-2023.)
Assertion
Ref Expression
bj-rcleq ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem bj-rcleq
StepHypRef Expression
1 nfcv 2920 . 2 𝑥𝐴
2 nfcv 2920 . 2 𝑥𝐵
3 nfcv 2920 . 2 𝑥𝑉
41, 2, 3bj-rcleqf 34743 1 ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1539   ∈ wcel 2112  ∀wral 3071   ∩ cin 3858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rab 3080  df-v 3412  df-in 3866 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator