Theorem eleq2w2 2730

Description: A weaker version of eleq2 2823 (but stronger than ax-9 2123 and elequ2 2128) that uses ax-12 2182 to avoid ax-8 2115 and df-clel 2809. Compare eleq2w 2818, 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 2727	. . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpi 216	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	19.21bi 2194	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541   ∈ wcel 2113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2726

This theorem is referenced by:  nfceqdf  2892  drnfc1  2916  drnfc2  2917  r1omhfb  35217  fineqvrep  35219  fineqvpow  35220  fineqvac  35221  r1omhfbregs  35242  fvineqsneu  37555  sge0f1o  46568  f1omoOLD  49081  discthing  49648