Theorem exan3 38276

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 38248	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐴))
2	1	el2v1 38204	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐴))
3		elecALTV 38248	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
4	3	el2v1 38204	. . 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 1919	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1776   ∈ wcel 2106  Vcvv 3478   class class class wbr 5148  [cec 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746

This theorem is referenced by:  brcoss2  38414