Theorem rabimbieq 38753

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 3419	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
4	1, 3	eqtri 2786	1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1561   ∈ wcel 2143  {crab 3415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-rab 3416

This theorem is referenced by:  abeqinbi  38755  dfsymrels4  39131  dfsymrels5  39132  dffunsALTV2  39269  dffunsALTV3  39270  dffunsALTV4  39271  dfdisjs2  39294  dfdisjs5  39297