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Theorem eqrrabd 4103
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 1913 . 2 𝑥𝜑
2 nfcv 2904 . 2 𝑥𝐵
3 nfrab1 3458 . 2 𝑥{𝑥𝐴𝜓}
4 eqrrabd.1 . . . . . 6 (𝜑𝐵𝐴)
54sseld 4001 . . . . 5 (𝜑 → (𝑥𝐵𝑥𝐴))
65pm4.71rd 562 . . . 4 (𝜑 → (𝑥𝐵 ↔ (𝑥𝐴𝑥𝐵)))
7 eqrrabd.2 . . . . 5 ((𝜑𝑥𝐴) → (𝑥𝐵𝜓))
87pm5.32da 578 . . . 4 (𝜑 → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝜓)))
96, 8bitrd 279 . . 3 (𝜑 → (𝑥𝐵 ↔ (𝑥𝐴𝜓)))
10 rabid 3459 . . 3 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
119, 10bitr4di 289 . 2 (𝜑 → (𝑥𝐵𝑥 ∈ {𝑥𝐴𝜓}))
121, 2, 3, 11eqrd 4022 1 (𝜑𝐵 = {𝑥𝐴𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2103  {crab 3438  wss 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-rab 3439  df-ss 3987
This theorem is referenced by:  usgrexmpl2nb0  47766  usgrexmpl2nb1  47767  usgrexmpl2nb2  47768  usgrexmpl2nb3  47769  usgrexmpl2nb4  47770  usgrexmpl2nb5  47771
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