Theorem exanres2

Description: Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021)

	Ref	Expression
Assertion	exanres2	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑢 ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝐵 ∈ [ 𝑢 ] 𝑅 ∧ 𝐶 ∈ [ 𝑢 ] 𝑆 ) ) )

Proof

Step	Hyp	Ref	Expression
1		exanres	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑢 ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ) )
2		exanres3	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑢 ∈ 𝐴 ( 𝐵 ∈ [ 𝑢 ] 𝑅 ∧ 𝐶 ∈ [ 𝑢 ] 𝑆 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝑢 𝑅 𝐵 ∧ 𝑢 𝑆 𝐶 ) ) )
3	1 2	bitr4d	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ∃ 𝑢 ( 𝑢 ( 𝑅 ↾ 𝐴 ) 𝐵 ∧ 𝑢 ( 𝑆 ↾ 𝐴 ) 𝐶 ) ↔ ∃ 𝑢 ∈ 𝐴 ( 𝐵 ∈ [ 𝑢 ] 𝑅 ∧ 𝐶 ∈ [ 𝑢 ] 𝑆 ) ) )