Theorem rabeqc 3427

Description: A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022.) (Proof shortened by SN, 15-Jan-2025.)

Hypothesis

Ref	Expression
rabeqc.1	⊢ (𝑥 ∈ 𝐴 → 𝜑)

Assertion

Ref	Expression
rabeqc	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabeqc

Step	Hyp	Ref	Expression
1		rabeqc.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝜑)
2	1	adantl 487	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → 𝜑)
3	2	rabeqcda 3426	. 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴)
4	3	mptru 1576	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ⊤wtru 1570   ∈ wcel 2146  {crab 3415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416

This theorem is used by:  rab0  4337  bday0  28209  2clwwlk2  30961  numclwwlk3lem2lem  30996  fply1  34143  vieta  34265  scottsn  35813  elnanelprv  36315  ipolub0  50245  ipoglb0  50247