Theorem raltpg

Description: Convert a restricted universal quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015)

	Ref	Expression
Hypotheses	ralprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ralprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
	raltpg.3	⊢ ( 𝑥 = 𝐶 → ( 𝜑 ↔ 𝜃 ) )
Assertion	raltpg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 , 𝐶 } 𝜑 ↔ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ralprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		ralprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
3		raltpg.3	⊢ ( 𝑥 = 𝐶 → ( 𝜑 ↔ 𝜃 ) )
4	1 2	ralprg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) )
5	3	ralsng	⊢ ( 𝐶 ∈ 𝑋 → ( ∀ 𝑥 ∈ { 𝐶 } 𝜑 ↔ 𝜃 ) )
6	4 5	bi2anan9	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ 𝐶 ∈ 𝑋 ) → ( ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐶 } 𝜑 ) ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) ) )
7	6	3impa	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐶 } 𝜑 ) ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) ) )
8		df-tp	⊢ { 𝐴 , 𝐵 , 𝐶 } = ( { 𝐴 , 𝐵 } ∪ { 𝐶 } )
9	8	raleqi	⊢ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 , 𝐶 } 𝜑 ↔ ∀ 𝑥 ∈ ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) 𝜑 )
10		ralunb	⊢ ( ∀ 𝑥 ∈ ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) 𝜑 ↔ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐶 } 𝜑 ) )
11	9 10	bitri	⊢ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 , 𝐶 } 𝜑 ↔ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∧ ∀ 𝑥 ∈ { 𝐶 } 𝜑 ) )
12		df-3an	⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ↔ ( ( 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) )
13	7 11 12	3bitr4g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 , 𝐶 } 𝜑 ↔ ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) ) )