Theorem rabbieq 35506

Description: Equivalent wff's correspond to restricted class abstractions which are equal with the same class. (Contributed by Peter Mazsa, 8-Jul-2019.)

Hypotheses

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

Assertion

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

Proof of Theorem rabbieq

Step	Hyp	Ref	Expression
1		rabbieq.1	. 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
2		rabbieq.2	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	rabbii 3473	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
4	1, 3	eqtri 2844	1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   ↔ wb 208   = wceq 1533  {crab 3142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-rab 3147

This theorem is referenced by:  dfrefrels3  35748  dfcnvrefrels3  35761  dfsymrels3  35776  refsymrels3  35796  dftrrels3  35806  dfeqvrels3  35818  dfdisjs3  35937  dfdisjs4  35938