Theorem eleq2w2 2722

Description: A weaker version of eleq2 2815 (but stronger than ax-9 2109 and elequ2 2114) that uses ax-12 2167 to avoid ax-8 2101 and df-clel 2803. Compare eleq2w 2810, 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 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eleq2w2

Step	Hyp	Ref	Expression
1		dfcleq 2719	. . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpi 215	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	19.21bi 2178	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   = wceq 1534   ∈ wcel 2099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-cleq 2718

This theorem is referenced by:  nfceqdf  2887  drnfc1  2912  drnfc2  2914  fineqvrep  34941  fineqvpow  34942  fineqvac  34943  f1omo  48261