Theorem rabeqc 3456

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 481	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → 𝜑)
3	2	rabeqcda 3455	. 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴)
4	3	mptru 1544	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

Syntax hints:   → wi 4   = wceq 1537  ⊤wtru 1538   ∈ wcel 2108  {crab 3443

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 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444

This theorem is referenced by:  bday0s  27891  2clwwlk2  30380  numclwwlk3lem2lem  30415  fply1  33549  elnanelprv  35397  ipolub0  48664  ipoglb0  48666