Theorem rabeqi 3428

Description: Equality theorem for restricted class abstractions. Inference form of rabeqf 3449. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Avoid ax-10 2175, ax-11 2191, ax-12 2212. (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 2854	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1i 635	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
4	3	rabbia2 3418	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜑}

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2142  {crab 3415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416

This theorem is used by:  f1ossf1o  7124  hsmex2  10423  iooval2  13411  fzval2  13544  phimullem  16844  pmtrsn  19595  dsmmbas2  21898  qtopres  23866  left1s  28099  right1s  28100  uvtxval  29748  cusgredg  29785  cffldtocusgr  29808  vtxdginducedm1  29904  finsumvtxdg2size  29911  konigsbergiedgw  30610  extwwlkfab  30714  zartopn  34274  satf0  35872  prjspeclsp  43372  k0004val0  44908  smflimlem4  47516  smfliminf  47573  isubgr0uhgr  48666  uspgrlimlem2  48782  uspgrlim  48785