Theorem rabbida

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019)

	Ref	Expression
Hypotheses	rabbida.n	⊢ Ⅎ 𝑥 𝜑
	rabbida.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	rabbida	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		rabbida.n	⊢ Ⅎ 𝑥 𝜑
2		rabbida.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 ↔ 𝜒 ) ) )
4	1 3	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜒 ) )
5		rabbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜒 ) ↔ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )
6	4 5	sylib	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )