Theorem rabimbieq 38826

Description: Restricted equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 22-Jul-2021.)

Hypotheses

Ref	Expression
rabimbieq.1	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
rabimbieq.2	⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rabimbieq	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

Proof of Theorem rabimbieq

Step	Hyp	Ref	Expression
1		rabimbieq.1	. 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
2		rabimbieq.2	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
3	2	rabbiia 3427	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
4	1, 3	eqtri 2792	1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  {crab 3423

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-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-rab 3424

This theorem is referenced by:  abeqinbi  38828  dfsymrels4  39204  dfsymrels5  39205  dffunsALTV2  39342  dffunsALTV3  39343  dffunsALTV4  39344  dfdisjs2  39367  dfdisjs5  39370