Theorem eleq2w2 2736

Description: A weaker version of eleq2 2833 (but stronger than ax-9 2118 and elequ2 2123) that uses ax-12 2178 to avoid ax-8 2110 and df-clel 2819. Compare eleq2w 2828, 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 2733	. . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpi 216	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	19.21bi 2190	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732

This theorem is referenced by:  nfceqdf  2904  drnfc1  2928  drnfc2  2930  fineqvrep  35071  fineqvpow  35072  fineqvac  35073  fvineqsneu  37377  f1omo  48574