Theorem rabbia2 2788

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
rabbia2.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))

Assertion

Ref	Expression
rabbia2	⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}

Proof of Theorem rabbia2

Step	Hyp	Ref	Expression
1		rabbia2.1	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))
2	1	a1i 9	. . 3 ⊢ (⊤ → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
3	2	rabbidva2 2787	. 2 ⊢ (⊤ → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
4	3	mptru 1407	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒}

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ⊤wtru 1399   ∈ wcel 2202  {crab 2515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-rab 2520

This theorem is referenced by:  sspw1or2  7446  clwwlknon2x  16359