Theorem bj-rabeqbida

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

	Ref	Expression
Hypotheses	bj-rabeqbida.nf	⊢ Ⅎ 𝑥 𝜑
	bj-rabeqbida.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	bj-rabeqbida.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	bj-rabeqbida	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		bj-rabeqbida.nf	⊢ Ⅎ 𝑥 𝜑
2		bj-rabeqbida.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		bj-rabeqbida.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
4	1 3	rabbida	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )
5	1 2	bj-rabeqd	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜒 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )
6	4 5	eqtrd	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜒 } )