Theorem bj-rabeqbid 37755

Description: Version of rabeqbidv 3429 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 3439	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
4		bj-rabeqbid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	1, 4	rabbid 3438	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
6	3, 5	eqtrd 2795	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  Ⅎwnf 1816  {crab 3412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413

This theorem is used by: (None)