Theorem exanres 38978

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

Assertion

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

Distinct variable groups:   𝑢,𝐵   𝑢,𝐶   𝑢,𝑉   𝑢,𝑊

Allowed substitution hints:   𝐴(𝑢)   𝑅(𝑢)   𝑆(𝑢)

Proof of Theorem exanres

Step	Hyp	Ref	Expression
1		brres 5984	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝑢(𝑅 ↾ 𝐴)𝐵 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵)))
2		brres 5984	. . . . 5 ⊢ (𝐶 ∈ 𝑊 → (𝑢(𝑆 ↾ 𝐴)𝐶 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶)))
3	1, 2	bi2anan9 649	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶))))
4		anandi 688	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶)))
5	3, 4	bitr4di 292	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))))
6	5	exbidv 1950	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))))
7		df-rex 3089	. 2 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
8	6, 7	bitr4di 292	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1808   ∈ wcel 2142  ∃wrex 3088   class class class wbr 5108   ↾ cres 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-res 5672

This theorem is used by:  exanres2  38980  br1cossres  39206