Theorem exanres 36357

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 5887	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝑢(𝑅 ↾ 𝐴)𝐵 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵)))
2		brres 5887	. . . . 5 ⊢ (𝐶 ∈ 𝑊 → (𝑢(𝑆 ↾ 𝐴)𝐶 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶)))
3	1, 2	bi2anan9 635	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶))))
4		anandi 672	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶)))
5	3, 4	bitr4di 288	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))))
6	5	exbidv 1925	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))))
7		df-rex 3069	. 2 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
8	6, 7	bitr4di 288	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064   class class class wbr 5070   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-res 5592

This theorem is referenced by:  exanres2  36359  br1cossres  36489