Theorem rabbida2 39317

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
rabbida2.1	⊢ Ⅎ𝑥𝜑
rabbida2.2	⊢ (𝜑 → 𝐴 = 𝐵)
rabbida2.3	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rabbida2	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Proof of Theorem rabbida2

Step	Hyp	Ref	Expression
1		rabbida2.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rabbida2.2	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	eleq2d 2687	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
4		rabbida2.3	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	3, 4	anbi12d 747	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
6	1, 5	abbid 2740	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)})
7		df-rab 2921	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
8		df-rab 2921	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}
9	6, 7, 8	3eqtr4g 2681	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  {cab 2608  {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921

This theorem is referenced by:  smflimmpt  41016