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Theorem eleq2w2 2728
Description: A weaker version of eleq2 2822 (but stronger than ax-9 2116 and elequ2 2121) that uses ax-12 2171 to avoid ax-8 2108 and df-clel 2810. Compare eleq2w 2817, whose setvars appear where the class variables are in this theorem, and vice versa. (Contributed by BJ, 24-Jun-2019.) Strengthen from setvar variables to class variables. (Revised by WL and SN, 23-Aug-2024.)
Assertion
Ref Expression
eleq2w2 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eleq2w2
StepHypRef Expression
1 dfcleq 2725 . . 3 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
21biimpi 215 . 2 (𝐴 = 𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
3219.21bi 2182 1 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-cleq 2724
This theorem is referenced by:  nfceqdf  2898  drnfc1  2922  drnfc2  2924  fineqvrep  34090  fineqvpow  34091  fineqvac  34092  f1omo  47517
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