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Mathbox for Giovanni Mascellani |
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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 3455 | . . . 4 ⊢ ∃𝑧 𝑧 = 𝐴 |
3 | ac6s6f.2 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
4 | vex 3444 | . . . . 5 ⊢ 𝑧 ∈ V | |
5 | ac6s6f.3 | . . . . 5 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
6 | 3, 4, 5 | ac6s6 35610 | . . . 4 ⊢ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) |
7 | 2, 6 | exan 1863 | . . 3 ⊢ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
8 | exdistr 1955 | . . 3 ⊢ (∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) ↔ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓))) | |
9 | 7, 8 | mpbir 234 | . 2 ⊢ ∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
10 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
11 | ac6s6f.4 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
12 | 10, 11 | raleqf 3350 | . . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓))) |
13 | 12 | biimpa 480 | . . 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 1913 | . 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 209 ∧ wa 399 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 Vcvv 3441 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 ax-ac2 9874 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-en 8493 df-r1 9177 df-rank 9178 df-card 9352 df-ac 9527 |
This theorem is referenced by: (None) |
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