Theorem rabeqcda 3427

Description: When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3623. (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 3074	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
2		rabeqcda.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	ex 412	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	pm4.71d 561	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
5	4	bicomd 222	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 ∈ 𝐴))
6	5	abbi1dv 2879	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = 𝐴)
7	1, 6	eqtrid 2791	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  {cab 2716  {crab 3069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074

This theorem is referenced by:  lrold  34056  prjcrv0  40450  mreclat  46235