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Theorem eleq2w2 2733
Description: A weaker version of eleq2 2830 (but stronger than ax-9 2118 and elequ2 2123) that uses ax-12 2177 to avoid ax-8 2110 and df-clel 2816. Compare eleq2w 2825, 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 2730 . . 3 (𝐴 = 𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
21biimpi 216 . 2 (𝐴 = 𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
3219.21bi 2189 1 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538   = wceq 1540  wcel 2108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729
This theorem is referenced by:  nfceqdf  2901  drnfc1  2925  drnfc2  2926  fineqvrep  35109  fineqvpow  35110  fineqvac  35111  fvineqsneu  37412  sge0f1o  46397  f1omo  48792
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