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Theorem ac4c 10494
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 3318 . . 3 (𝑦 = 𝐴 → (∀𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥) ↔ ∀𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
32exbidv 1917 . 2 (𝑦 = 𝐴 → (∃𝑓𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥) ↔ ∃𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)))
4 ac4 10493 . 2 𝑓𝑥𝑦 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)
51, 3, 4vtocl 3542 1 𝑓𝑥𝐴 (𝑥 ≠ ∅ → (𝑓𝑥) ∈ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wex 1774  wcel 2099  wne 2936  wral 3057  Vcvv 3470  c0 4319  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-ac2 10481
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ac 10134
This theorem is referenced by:  axdclem2  10538
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