Theorem exanres3 38611

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 38580	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
2	1	el2v1 38538	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝐵))
3		elecALTV 38580	. . . 4 ⊢ ((𝑢 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ [𝑢]𝑆 ↔ 𝑢𝑆𝐶))
4	3	el2v1 38538	. . 3 ⊢ (𝐶 ∈ 𝑊 → (𝐶 ∈ [𝑢]𝑆 ↔ 𝑢𝑆𝐶))
5	2, 4	bi2anan9 639	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
6	5	rexbidv 3159	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∃wrex 3059  Vcvv 3427   class class class wbr 5074  [cec 8630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ec 8634

This theorem is referenced by:  exanres2  38612  br1cossres2  38839