Theorem rabbid 3440

Description: Version of rabbidv 3420 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 484	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	rabbida 3439	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559  Ⅎwnf 1802   ∈ wcel 2141  {crab 3413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-rab 3414

This theorem is referenced by:  satfv1  35677  bj-rabeqbid  37370  bj-seex  37371