Theorem rabbida3 40822

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

Hypotheses

Ref	Expression
rabbida3.1	⊢ Ⅎ𝑥𝜑
rabbida3.2	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))

Assertion

Ref	Expression
rabbida3	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Proof of Theorem rabbida3

Step	Hyp	Ref	Expression
1		rabbida3.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		rabbida3.2	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
3	1, 2	abbid 2846	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)})
4		df-rab 3098	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
5		df-rab 3098	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}
6	3, 4, 5	3eqtr4g 2840	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507  Ⅎwnf 1746   ∈ wcel 2050  {cab 2759  {crab 3093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-9 2059  ax-12 2106  ax-ext 2751

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-rab 3098

This theorem is referenced by:  smflimmpt  42513  smflimsupmpt  42532  smfliminfmpt  42535