Theorem exanres3 35552

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 35526	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
2	1	el2v1 35489	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
3		elecALTV 35526	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ [𝑢]𝑆 ↔ 𝑢𝑆𝐶))
4	3	el2v1 35489	. . 3 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ [𝑢]𝑆 ↔ 𝑢𝑆𝐶))
5	2, 4	bi2anan9 637	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
6	5	rexbidv 3297	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  ∃wrex 3139  Vcvv 3494   class class class wbr 5065  [cec 8286

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ec 8290

This theorem is referenced by:  exanres2  35553  br1cossres2  35684