Theorem rabeqc 3426

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 485	. . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → 𝜑)
3	2	rabeqcda 3425	. 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴)
4	3	mptru 1567	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴

Syntax hints:   → wi 4   = wceq 1560  ⊤wtru 1561   ∈ wcel 2142  {crab 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415

This theorem is referenced by:  rab0  4339  bday0  27904  2clwwlk2  30550  numclwwlk3lem2lem  30585  fply1  33754  vieta  33877  elnanelprv  35779  ipolub0  49613  ipoglb0  49615