Theorem exan3 38804

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 38775	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐴))
2	1	el2v1 38733	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐴))
3		elecALTV 38775	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
4	3	el2v1 38733	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
5	2, 4	bi2anan9 647	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))
6	5	exbidv 1943	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∃wex 1801   ∈ wcel 2144  Vcvv 3456   class class class wbr 5102  [cec 8678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682

This theorem is referenced by:  brcoss2  39026