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Theorem eleq2w2 2731
Description: A weaker version of eleq2 2828 (but stronger than ax-9 2116 and elequ2 2121) that uses ax-12 2175 to avoid ax-8 2108 and df-clel 2814. Compare eleq2w 2823, 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 2728 . . 3 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
21biimpi 216 . 2 (𝐴 = 𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
3219.21bi 2187 1 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727
This theorem is referenced by:  nfceqdf  2899  drnfc1  2923  drnfc2  2924  fineqvrep  35088  fineqvpow  35089  fineqvac  35090  fvineqsneu  37394  sge0f1o  46338  f1omo  48691
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