Theorem rabeq12f

Description: Equality deduction for restricted class abstraction. (Contributed by Giovanni Mascellani, 10-Apr-2018)

	Ref	Expression
Hypotheses	rabeq12f.1	⊢ Ⅎ 𝑥 𝐴
	rabeq12f.2	⊢ Ⅎ 𝑥 𝐵
Assertion	rabeq12f	⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝜓 ) ) → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } )

Proof

Step	Hyp	Ref	Expression
1		rabeq12f.1	⊢ Ⅎ 𝑥 𝐴
2		rabeq12f.2	⊢ Ⅎ 𝑥 𝐵
3		rabbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝜓 ) ↔ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐴 ∣ 𝜓 } )
4	3	biimpi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝜓 ) → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐴 ∣ 𝜓 } )
5	1 2	rabeqf	⊢ ( 𝐴 = 𝐵 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } )
6	4 5	sylan9eqr	⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝜑 ↔ 𝜓 ) ) → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∈ 𝐵 ∣ 𝜓 } )