Theorem rabbida4 3439

Description: Version of rabbidva2 3416 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 2830	. 2 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)})
4		df-rab 3415	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
5		df-rab 3415	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜒} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜒)}
6	3, 4, 5	3eqtr4g 2822	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560  Ⅎwnf 1803   ∈ wcel 2142  {cab 2740  {crab 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-rab 3415

This theorem is referenced by:  rabbida  3440  rabeqd  3442  rabeqf  3448