Theorem exan3 38282

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 38255	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐴))
2	1	el2v1 38211	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐴))
3		elecALTV 38255	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
4	3	el2v1 38211	. . 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 1921	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  Vcvv 3447   class class class wbr 5107  [cec 8669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673

This theorem is referenced by:  brcoss2  38423