Theorem exanres 38640

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 5947	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝑢(𝑅 ↾ 𝐴)𝐵 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵)))
2		brres 5947	. . . . 5 ⊢ (𝐶 ∈ 𝑊 → (𝑢(𝑆 ↾ 𝐴)𝐶 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶)))
3	1, 2	bi2anan9 639	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝐵) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑆𝐶))))
4		anandi 677	. . . 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 1923	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))))
7		df-rex 3063	. 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 1781   ∈ wcel 2114  ∃wrex 3062   class class class wbr 5086   ↾ cres 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5632  df-res 5638

This theorem is referenced by:  exanres2  38642  br1cossres  38868