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Theorem exanres2 36690
Description: Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.)
Assertion
Ref Expression
exanres2 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆)))
Distinct variable groups:   𝑢,𝐵   𝑢,𝐶   𝑢,𝑉   𝑢,𝑊
Allowed substitution hints:   𝐴(𝑢)   𝑅(𝑢)   𝑆(𝑢)

Proof of Theorem exanres2
StepHypRef Expression
1 exanres 36688 . 2 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
2 exanres3 36689 . 2 ((𝐵𝑉𝐶𝑊) → (∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑆𝐶)))
31, 2bitr4d 282 1 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑆𝐴)𝐶) ↔ ∃𝑢𝐴 (𝐵 ∈ [𝑢]𝑅𝐶 ∈ [𝑢]𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wex 1782  wcel 2107  wrex 3072   class class class wbr 5104  cres 5634  [cec 8605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8609
This theorem is referenced by: (None)
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