Theorem rabbida4 3455

Description: Version of rabbidva2 3432 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)

Hypotheses

Ref	Expression
rabbida4.nf	⊢ Ⅎ𝑥𝜑
rabbida4.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))

Assertion

Ref	Expression
rabbida4	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Proof of Theorem rabbida4

Step	Hyp	Ref	Expression
1		rabbida4.nf	. . 3 ⊢ Ⅎ𝑥𝜑
2		rabbida4.1	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
3	1, 2	abbid 2801	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)})
4		df-rab 3431	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
5		df-rab 3431	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}
6	3, 4, 5	3eqtr4g 2795	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539  Ⅎwnf 1783   ∈ wcel 2104  {cab 2707  {crab 3430

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 1911  ax-6 1969  ax-7 2009  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-rab 3431

This theorem is referenced by:  rabbida  3456  rabeqd  3458