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Metamath Proof Explorer

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 GG, 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 ) ) )