Theorem bj-rabeqbid 35108

Description: Version of rabeqbidv 3420 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	bj-rabeqd 35107	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
4		bj-rabeqbid.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	1, 4	rabbid 3410	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
6	3, 5	eqtrd 2778	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  Ⅎwnf 1786  {crab 3068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073

This theorem is referenced by: (None)