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Theorem cpcoll2d 44255
Description: cpcolld 44254 with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023.)
Hypotheses
Ref Expression
cpcoll2d.1 (𝜑𝑥𝐴)
cpcoll2d.2 (𝜑 → ∃𝑦 𝑥𝐹𝑦)
Assertion
Ref Expression
cpcoll2d (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem cpcoll2d
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 cpcoll2d.2 . . 3 (𝜑 → ∃𝑦 𝑥𝐹𝑦)
2 breq2 5152 . . . 4 (𝑎 = 𝑦 → (𝑥𝐹𝑎𝑥𝐹𝑦))
32cbvexvw 2034 . . 3 (∃𝑎 𝑥𝐹𝑎 ↔ ∃𝑦 𝑥𝐹𝑦)
41, 3sylibr 234 . 2 (𝜑 → ∃𝑎 𝑥𝐹𝑎)
5 cpcoll2d.1 . . . . 5 (𝜑𝑥𝐴)
65adantr 480 . . . 4 ((𝜑𝑥𝐹𝑎) → 𝑥𝐴)
7 simpr 484 . . . 4 ((𝜑𝑥𝐹𝑎) → 𝑥𝐹𝑎)
86, 7cpcolld 44254 . . 3 ((𝜑𝑥𝐹𝑎) → ∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎)
92cbvrexvw 3236 . . 3 (∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎 ↔ ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
108, 9sylib 218 . 2 ((𝜑𝑥𝐹𝑎) → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
114, 10exlimddv 1933 1 (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wcel 2106  wrex 3068   class class class wbr 5148   Coll ccoll 44246
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-r1 9802  df-rank 9803  df-scott 44232  df-coll 44247
This theorem is referenced by:  grumnudlem  44281
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