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Theorem bj-rcleq 36413
Description: Relative version of dfcleq 2719. (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 2897 . 2 𝑥𝐴
2 nfcv 2897 . 2 𝑥𝐵
3 nfcv 2897 . 2 𝑥𝑉
41, 2, 3bj-rcleqf 36412 1 ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  wral 3055  cin 3942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rab 3427  df-v 3470  df-in 3950
This theorem is referenced by: (None)
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