Theorem rabeqcda 3432

Description: When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3433. (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 3421	. 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 2869	. 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
6	1, 5	eqtr4id 2790	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2714  {crab 3420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421

This theorem is referenced by:  rabeqc  3433  cmnbascntr  19791  lrold  27865  unitscyglem4  42216  prjcrv0  42623  isubgrvtxuhgr  47844  stgrnbgr0  47943  mreclat  48938