Theorem exanres3 38354

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 38323	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
2	1	el2v1 38284	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
3		elecALTV 38323	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ [𝑢]𝑆 ↔ 𝑢𝑆𝐶))
4	3	el2v1 38284	. . 3 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ [𝑢]𝑆 ↔ 𝑢𝑆𝐶))
5	2, 4	bi2anan9 638	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
6	5	rexbidv 3157	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  ∃wrex 3057  Vcvv 3437   class class class wbr 5093  [cec 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ec 8630

This theorem is referenced by:  exanres2  38355  br1cossres2  38562