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Theorem eqrrabd 30276
Description: Deduce equality with a restricted abstraction. (Contributed by Thierry Arnoux, 11-Apr-2024.)
Hypotheses
Ref Expression
eqrrabd.1 (𝜑𝐵𝐴)
eqrrabd.2 ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))
Assertion
Ref Expression
eqrrabd (𝜑𝐵 = {𝑥𝐴𝜓})
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem eqrrabd
StepHypRef Expression
1 nfv 1915 . 2 𝑥𝜑
2 nfcv 2958 . 2 𝑥𝐵
3 nfrab1 3340 . 2 𝑥{𝑥𝐴𝜓}
4 eqrrabd.1 . . . . . 6 (𝜑𝐵𝐴)
54sseld 3917 . . . . 5 (𝜑 → (𝑥𝐵𝑥𝐴))
65pm4.71rd 566 . . . 4 (𝜑 → (𝑥𝐵 ↔ (𝑥𝐴𝑥𝐵)))
7 eqrrabd.2 . . . . 5 ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))
87pm5.32da 582 . . . 4 (𝜑 → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝜓)))
96, 8bitrd 282 . . 3 (𝜑 → (𝑥𝐵 ↔ (𝑥𝐴𝜓)))
10 rabid 3334 . . 3 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
119, 10syl6bbr 292 . 2 (𝜑 → (𝑥𝐵𝑥 ∈ {𝑥𝐴𝜓}))
121, 2, 3, 11eqrd 3937 1 (𝜑𝐵 = {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {crab 3113  wss 3884
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901
This theorem is referenced by: (None)
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