Theorem rabimbieq 37855

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 3422	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
4	1, 3	eqtri 2753	1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {crab 3418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-rab 3419

This theorem is referenced by:  abeqinbi  37857  dfsymrels4  38151  dfsymrels5  38152  dffunsALTV2  38288  dffunsALTV3  38289  dffunsALTV4  38290  dfdisjs2  38313  dfdisjs5  38316