Theorem exanres3 37160

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

Assertion

Ref	Expression
exanres3	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ ∃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 exanres3

Step	Hyp	Ref	Expression
1		elecALTV 37129	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
2	1	el2v1 37080	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
3		elecALTV 37129	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ [𝑢]𝑆 ↔ 𝑢𝑆𝐶))
4	3	el2v1 37080	. . 3 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ [𝑢]𝑆 ↔ 𝑢𝑆𝐶))
5	2, 4	bi2anan9 637	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
6	5	rexbidv 3178	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474   class class class wbr 5148  [cec 8700

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8704

This theorem is referenced by:  exanres2  37161  br1cossres2  37305