Theorem rabbid 3439

Description: Version of rabbidv 3419 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)

Hypotheses

Ref	Expression
rabbid.n	⊢ Ⅎ𝑥𝜑
rabbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem rabbid

Step	Hyp	Ref	Expression
1		rabbid.n	. 2 ⊢ Ⅎ𝑥𝜑
2		rabbid.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rabbida 3438	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  {crab 3411

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 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-rab 3412

This theorem is referenced by:  satfv1  35352  bj-rabeqbid  36906  bj-seex  36907