Theorem rabbieq 3398

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 3395	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
4	1, 3	eqtri 2760	1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   ↔ wb 206   = wceq 1542  {crab 3390

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-rab 3391

This theorem is referenced by:  dfdif3  4058  elneldisj  4333  elnelun  4334  1arithufd  33623  dfrefrels3  38929  dfcnvrefrels3  38944  dfsymrels3  38961  refsymrels3  38985  dftrrels3  38995  dfeqvrels3  39008  dfdisjs3  39130  dfdisjs4  39131  isubgr0uhgr  48361  grlimedgclnbgr  48483