Theorem rabbid 3419

Description: Version of rabbidv 3399 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 481	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rabbida 3418	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  {crab 3392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-rab 3393

This theorem is referenced by:  satfv1  35598  bj-rabeqbid  37281  bj-seex  37282