Theorem bj-rabbid 32107

Description: Version of rabbidv 3164 with dv condition replaced by non-freeness hypothesis. (Contributed by BJ, 27-Apr-2019.)

Hypotheses

Ref	Expression
bj-rabbid.nf	⊢ Ⅎ𝑥𝜑
bj-rabbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem bj-rabbid

Step	Hyp	Ref	Expression
1		bj-rabbid.nf	. 2 ⊢ Ⅎ𝑥𝜑
2		bj-rabbid.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	bj-rabbida 32106	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  {crab 2900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rab 2905

This theorem is referenced by:  bj-rabeqbid  32109  bj-seex  32111