Theorem eleq2w2 2733

Description: A weaker version of eleq2 2826 (but stronger than ax-9 2124 and elequ2 2129) that uses ax-12 2185 to avoid ax-8 2116 and df-clel 2812. Compare eleq2w 2821, 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 2730	. . 3 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
2	1	biimpi 216	. 2 ⊢ (𝐴 = 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	2	19.21bi 2197	1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729

This theorem is referenced by:  nfceqdf  2895  drnfc1  2919  drnfc2  2920  r1omhfb  35290  fineqvrep  35292  fineqvpow  35293  fineqvac  35294  r1omhfbregs  35315  fvineqsneu  37666  sge0f1o  46740  f1omoOLD  49253  discthing  49820