Theorem rabbid 3404

Description: Version of rabbidv 3409 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 3403	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2107  {crab 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2715  df-cleq 2729  df-ral 3067  df-rab 3071

This theorem is referenced by:  satfv1  33267  bj-rabeqbid  35077  bj-seex  35079