Theorem rabbidvaOLD

Description: Obsolete proof of rabbidva as of 4-Dec-2023. (Contributed by NM, 28-Nov-2003) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	rabbidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	rabbidvaOLD	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		rabbidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
2	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜒 ) )
3		rabbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 ↔ 𝜒 ) ↔ { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )
4	2 3	sylib	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )