Theorem ralprgOLD

Description: Obsolete version of ralprg as of 30-Sep-2024. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015) (Proof shortened by AV, 8-Apr-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ralprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ralprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
Assertion	ralprgOLD	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ralprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		ralprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
3		nfv	⊢ Ⅎ 𝑥 𝜓
4		nfv	⊢ Ⅎ 𝑥 𝜒
5	3 4 1 2	ralprgf	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )