Theorem ralprg

Description: Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 30-Sep-2024)

	Ref	Expression
Hypotheses	ralprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ralprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
Assertion	ralprg	⊢ ( ( 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		df-pr	⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
4	3	raleqi	⊢ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ∀ 𝑥 ∈ ( { 𝐴 } ∪ { 𝐵 } ) 𝜑 )
5		ralunb	⊢ ( ∀ 𝑥 ∈ ( { 𝐴 } ∪ { 𝐵 } ) 𝜑 ↔ ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐵 } 𝜑 ) )
6	4 5	bitri	⊢ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐵 } 𝜑 ) )
7	1	ralsng	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )
8	2	ralsng	⊢ ( 𝐵 ∈ 𝑊 → ( ∀ 𝑥 ∈ { 𝐵 } 𝜑 ↔ 𝜒 ) )
9	7 8	bi2anan9	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( ∀ 𝑥 ∈ { 𝐴 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐵 } 𝜑 ) ↔ ( 𝜓 ∧ 𝜒 ) ) )
10	6 9	bitrid	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )