Theorem rabbieq 3423

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 3420	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
4	1, 3	eqtri 2786	1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  {crab 3415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-rab 3416

This theorem is used by:  dfdif3  4057  elneldisj  4344  elnelun  4345  1arithufd  34133  mirimanoffr  36581  dfrefrels3  39666  dfcnvrefrels3  39681  dfsymrels3  39698  refsymrels3  39722  dftrrels3  39732  dfeqvrels3  39745  dfdisjs3  39867  dfdisjs4  39868  isubgr0uhgr  49110  grlimedgclnbgr  49232