Theorem rabbidaOLD

Description: Obsolete version of rabbida as of 14-Mar-2025. (Contributed by BJ, 27-Apr-2019) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		rabbidaOLD.n	⊢ Ⅎ 𝑥 𝜑
2		rabbidaOLD.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 } )