Theorem rabeqcda 3408

Description: When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3409. (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 3398	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
2		rabeqcda.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	ex 412	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	pm4.71d 561	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
5	4	eqabdv 2867	. 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
6	1, 5	eqtr4id 2788	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2712  {crab 3397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398

This theorem is referenced by:  rabeqc  3409  chnfi  18555  cmnbascntr  19732  lrold  27869  unitscyglem4  42391  prjcrv0  42818  isubgrvtxuhgr  48052  stgrnbgr0  48152  mreclat  49184