Theorem rabbieq 3397

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 3394	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
4	1, 3	eqtri 2759	1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   ↔ wb 206   = wceq 1542  {crab 3389

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-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-rab 3390

This theorem is referenced by:  dfdif3  4057  elneldisj  4332  elnelun  4333  1arithufd  33608  dfrefrels3  38915  dfcnvrefrels3  38930  dfsymrels3  38947  refsymrels3  38971  dftrrels3  38981  dfeqvrels3  38994  dfdisjs3  39116  dfdisjs4  39117  isubgr0uhgr  48349  grlimedgclnbgr  48471