Theorem rabeqi 3425

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 3445. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2178, ax-11 2194, ax-12 2213. (Revised by GG, 3-Jun-2024.)

Hypothesis

Ref	Expression
rabeqi.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
rabeqi	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

Proof of Theorem rabeqi

Step	Hyp	Ref	Expression
1		rabeqi.1	. . . 4 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2852	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1i 636	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
4	3	rabbia2 3415	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {crab 3412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413

This theorem is used by:  f1ossf1o  7122  hsmex2  10435  iooval2  13431  fzval2  13564  phimullem  16870  pmtrsn  19646  dsmmbas2  21950  qtopres  23924  left1s  28160  right1s  28161  uvtxval  29847  cusgredg  29884  cffldtocusgr  29907  vtxdginducedm1  30003  finsumvtxdg2size  30010  konigsbergiedgw  30728  extwwlkfab  30832  zartopn  34385  satf0  35951  prjspeclsp  43458  k0004val0  44994  smflimlem4  47602  smfliminf  47659  isubgr0uhgr  48789  uspgrlimlem2  48905  uspgrlim  48908