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Metamath Proof Explorer

Theorem exanres

Description: Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 2-May-2021)

Ref Expression
Assertion exanres ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑢 ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 brres ⊢ ( 𝐵 ∈ 𝑉 → ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ↔ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑅 𝐵 ) ) )
2 brres ⊢ ( 𝐶 ∈ 𝑊 → ( 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ↔ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑆 𝐶 ) ) )
3 1 2 bi2anan9 ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ) ↔ ( ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑆 𝐶 ) ) ) )
4 anandi ⊢ ( ( 𝑢 ∈ 𝐴 ∧ ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ) ↔ ( ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑅 𝐵 ) ∧ ( 𝑢 ∈ 𝐴 ∧ 𝑢 𝑆 𝐶 ) ) )
5 3 4 bitr4di ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ) ↔ ( 𝑢 ∈ 𝐴 ∧ ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ) ) )
6 5 exbidv ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑢 ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ) ↔ ∃ 𝑢 ( 𝑢 ∈ 𝐴 ∧ ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ) ) )
7 df-rex ⊢ ( ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ↔ ∃ 𝑢 ( 𝑢 ∈ 𝐴 ∧ ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ) )
8 6 7 bitr4di ⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑢 ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ) )