Theorem rabeqcda 3412

Description: When 𝜓 is always true in a context, a restricted class abstraction is equal to the restricting class. Deduction form of rabeqc 3413. (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 3402	. 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 2870	. 2 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
6	1, 5	eqtr4id 2791	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  {crab 3401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402

This theorem is referenced by:  rabeqc  3413  chnfi  18569  cmnbascntr  19749  lrold  27908  unitscyglem4  42572  prjcrv0  42995  isubgrvtxuhgr  48228  stgrnbgr0  48328  mreclat  49360