Theorem exanres 38251

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 6016	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝑢(𝑅 ↾ 𝐴)𝐵 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵)))
2		brres 6016	. . . . 5 ⊢ (𝐶 ∈ 𝑊 → (𝑢(𝑆 ↾ 𝐴)𝐶 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶)))
3	1, 2	bi2anan9 637	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶))))
4		anandi 675	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶)))
5	3, 4	bitr4di 289	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))))
6	5	exbidv 1920	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))))
7		df-rex 3077	. 2 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
8	6, 7	bitr4di 289	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076   class class class wbr 5166   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-res 5712

This theorem is referenced by:  exanres2  38253  br1cossres  38395