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Metamath Proof Explorer

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 ) ) )