Theorem eleq2w2 2759

Description: A weaker version of eleq2 2852 (but stronger than ax-9 2153 and elequ2 2158) that uses ax-12 2213 to avoid ax-8 2145 and df-clel 2838. Compare eleq2w 2847, 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 2756	. . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpi 219	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	19.21bi 2225	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570   ∈ wcel 2143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755

This theorem is referenced by:  nfceqdf  2921  drnfc1  2944  drnfc2  2945  plngval  29059  prlngplngtr  29209  r1omhfb  35508  fineqvrep  35527  fineqvpow  35528  fineqvac  35529  r1omhfbregs  35550  fvineqsneu  38057  sge0f1o  47096  f1omoOLD  49672  discthing  50239