Theorem rabeqcda 3433

Description: When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3434. (Contributed by Steven Nguyen, 7-Jun-2023.)

Hypothesis

Ref	Expression
rabeqcda.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)

Assertion

Ref	Expression
rabeqcda	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rabeqcda

Step	Hyp	Ref	Expression
1		df-rab 3423	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
2		rabeqcda.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	ex 417	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	pm4.71d 570	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
5	4	eqabdv 2902	. 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
6	1, 5	eqtr4id 2823	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  {crab 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3423

This theorem is referenced by:  rabeqc  3434  chnfi  18690  cmnbascntr  19875  lrold  28056  0mplrim  33849  unitscyglem4  42890  prjcrv0  43292  isubgrvtxuhgr  48553  stgrnbgr0  48653  mreclat  49695