Theorem rabeqi 3429

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 3449. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2180, ax-11 2197, ax-12 2216. (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 2856	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1i 636	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
4	3	rabbia2 3419	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {crab 3416

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417

This theorem is used by:  f1ossf1o  7153  hsmex2  10538  iooval2  13536  fzval2  13669  phimullem  16985  pmtrsn  19763  dsmmbas2  22073  qtopres  24047  left1s  28311  right1s  28312  uvtxval  29998  cusgredg  30035  cffldtocusgr  30058  vtxdginducedm1  30154  finsumvtxdg2size  30161  konigsbergiedgw  30879  extwwlkfab  30983  zartopn  34577  satf0  36304  prjspeclsp  43833  k0004val0  45346  smflimlem4  47989  smfliminf  48046  isubgr0uhgr  49176  uspgrlimlem2  49292  uspgrlim  49295