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Theorem ac4c 9245
Description: Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997.)
Hypothesis
Ref Expression
ac4c.1 𝐴 ∈ V
Assertion
Ref Expression
ac4c 𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)
Distinct variable group:   𝑥,𝑓,𝐴

Proof of Theorem ac4c
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ac4c.1 . 2 𝐴 ∈ V
2 raleq 3127 . . 3 (𝑦 = 𝐴 → (∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥) ↔ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
32exbidv 1847 . 2 (𝑦 = 𝐴 → (∃𝑓𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥) ↔ ∃𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
4 ac4 9244 . 2 𝑓𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)
51, 3, 4vtocl 3245 1 𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  Vcvv 3186  c0 3893  cfv 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-ac2 9232
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ac 8886
This theorem is referenced by:  axdclem2  9289
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