Theorem exan3 38296

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

Assertion

Ref	Expression
exan3	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑉   𝑢,𝑊

Allowed substitution hint:   𝑅(𝑢)

Proof of Theorem exan3

Step	Hyp	Ref	Expression
1		elecALTV 38268	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐴))
2	1	el2v1 38225	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐴))
3		elecALTV 38268	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
4	3	el2v1 38225	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
5	2, 4	bi2anan9 638	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
6	5	exbidv 1920	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1778   ∈ wcel 2107  Vcvv 3479   class class class wbr 5142  [cec 8744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748

This theorem is referenced by:  brcoss2  38434