Theorem bj-rabeqbid 36860

Description: Version of rabeqbidv 3432 with two disjoint variable conditions removed and the third replaced by a nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019.)

Hypotheses

Ref	Expression
bj-rabeqbid.nf	⊢ Ⅎ𝑥𝜑
bj-rabeqbid.1	⊢ (𝜑 → 𝐴 = 𝐵)
bj-rabeqbid.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem bj-rabeqbid

Step	Hyp	Ref	Expression
1		bj-rabeqbid.nf	. . 3 ⊢ Ⅎ𝑥𝜑
2		bj-rabeqbid.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	1, 2	rabeqd 3442	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
4		bj-rabeqbid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	1, 4	rabbid 3441	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
6	3, 5	eqtrd 2769	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539  Ⅎwnf 1782  {crab 3413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3414

This theorem is referenced by: (None)