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Metamath Proof Explorer

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 ) ) )