Theorem ralprgf

Description: Convert a restricted universal quantification over a pair to a conjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011) (Revised by AV, 8-Apr-2023)

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

Proof

Step	Hyp	Ref	Expression
1		ralprgf.1	⊢ Ⅎ 𝑥 𝜓
2		ralprgf.2	⊢ Ⅎ 𝑥 𝜒
3		ralprgf.a	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4		ralprgf.b	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
5		df-pr	⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
6	5	raleqi	⊢ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ∀ 𝑥 ∈ ( { 𝐴 } ∪ { 𝐵 } ) 𝜑 )
7		ralunb	⊢ ( ∀ 𝑥 ∈ ( { 𝐴 } ∪ { 𝐵 } ) 𝜑 ↔ ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐵 } 𝜑 ) )
8	6 7	bitri	⊢ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐵 } 𝜑 ) )
9	1 3	ralsngf	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )
10	2 4	ralsngf	⊢ ( 𝐵 ∈ 𝑊 → ( ∀ 𝑥 ∈ { 𝐵 } 𝜑 ↔ 𝜒 ) )
11	9 10	bi2anan9	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐵 } 𝜑 ) ↔ ( 𝜓 ∧ 𝜒 ) ) )
12	8 11	bitrid	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )