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Theorem cpcoll2d 40887
 Description: cpcolld 40886 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 5056 . . . 4 (𝑎 = 𝑦 → (𝑥𝐹𝑎𝑥𝐹𝑦))
32cbvexvw 2045 . . 3 (∃𝑎 𝑥𝐹𝑎 ↔ ∃𝑦 𝑥𝐹𝑦)
41, 3sylibr 237 . 2 (𝜑 → ∃𝑎 𝑥𝐹𝑎)
5 cpcoll2d.1 . . . . 5 (𝜑𝑥𝐴)
65adantr 484 . . . 4 ((𝜑𝑥𝐹𝑎) → 𝑥𝐴)
7 simpr 488 . . . 4 ((𝜑𝑥𝐹𝑎) → 𝑥𝐹𝑎)
86, 7cpcolld 40886 . . 3 ((𝜑𝑥𝐹𝑎) → ∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎)
92cbvrexvw 3435 . . 3 (∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎 ↔ ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
108, 9sylib 221 . 2 ((𝜑𝑥𝐹𝑎) → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
114, 10exlimddv 1937 1 (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2115  ∃wrex 3134   class class class wbr 5052   Coll ccoll 40878 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-r1 9190  df-rank 9191  df-scott 40864  df-coll 40879 This theorem is referenced by:  grumnudlem  40913
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