Theorem rabbieq 3429

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 3426	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ 𝜓}
4	1, 3	eqtri 2759	1 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜓}

Syntax hints:   ↔ wb 206   = wceq 1540  {crab 3420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-rab 3421

This theorem is referenced by:  dfdif3  4097  1arithufd  33568  dfrefrels3  38537  dfcnvrefrels3  38552  dfsymrels3  38569  refsymrels3  38589  dftrrels3  38599  dfeqvrels3  38612  dfdisjs3  38733  dfdisjs4  38734  isubgr0uhgr  47853