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Mirrors > Home > MPE Home > Th. List > Mathboxes > ac6s6f | Structured version Visualization version GIF version |
Description: Generalization of the Axiom of Choice to classes, moving the existence condition in the consequent. (Contributed by Giovanni Mascellani, 20-Aug-2018.) |
Ref | Expression |
---|---|
ac6s6f.1 | ⊢ 𝐴 ∈ V |
ac6s6f.2 | ⊢ Ⅎ𝑦𝜓 |
ac6s6f.3 | ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) |
ac6s6f.4 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
ac6s6f | ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac6s6f.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | 1 | isseti 3456 | . . . 4 ⊢ ∃𝑧 𝑧 = 𝐴 |
3 | ac6s6f.2 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
4 | vex 3445 | . . . . 5 ⊢ 𝑧 ∈ V | |
5 | ac6s6f.3 | . . . . 5 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
6 | 3, 4, 5 | ac6s6 36435 | . . . 4 ⊢ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) |
7 | 2, 6 | exan 1864 | . . 3 ⊢ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
8 | exdistr 1957 | . . 3 ⊢ (∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) ↔ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓))) | |
9 | 7, 8 | mpbir 230 | . 2 ⊢ ∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
10 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
11 | ac6s6f.4 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
12 | 10, 11 | raleqf 3324 | . . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓))) |
13 | 12 | biimpa 477 | . . 3 ⊢ ((𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) → ∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) |
14 | 13 | 2eximi 1837 | . 2 ⊢ (∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) → ∃𝑧∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) |
15 | ax5e 1914 | . 2 ⊢ (∃𝑧∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓) → ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) | |
16 | 9, 14, 15 | mp2b 10 | 1 ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2105 Ⅎwnfc 2884 ∀wral 3061 Vcvv 3441 ‘cfv 6479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-reg 9449 ax-inf2 9498 ax-ac2 10320 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-en 8805 df-r1 9621 df-rank 9622 df-card 9796 df-ac 9973 |
This theorem is referenced by: (None) |
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