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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 | . . . . . 6 ⊢ 𝐴 ∈ V | |
2 | 1 | isseti 3240 | . . . . 5 ⊢ ∃𝑧 𝑧 = 𝐴 |
3 | ac6s6f.2 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
4 | vex 3234 | . . . . . 6 ⊢ 𝑧 ∈ V | |
5 | ac6s6f.3 | . . . . . 6 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
6 | 3, 4, 5 | ac6s6 34110 | . . . . 5 ⊢ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) |
7 | 2, 6 | pm3.2i 470 | . . . 4 ⊢ (∃𝑧 𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
8 | 7 | exan 1828 | . . 3 ⊢ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
9 | exdistr 1922 | . . 3 ⊢ (∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) ↔ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓))) | |
10 | 8, 9 | mpbir 221 | . 2 ⊢ ∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
11 | nfcv 2793 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
12 | ac6s6f.4 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
13 | 11, 12 | raleqf 3164 | . . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓))) |
14 | 13 | biimpa 500 | . . 3 ⊢ ((𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) → ∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) |
15 | 14 | 2eximi 1803 | . 2 ⊢ (∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) → ∃𝑧∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) |
16 | ax5e 1881 | . 2 ⊢ (∃𝑧∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓) → ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) | |
17 | 10, 15, 16 | mp2b 10 | 1 ⊢ ∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∃wex 1744 Ⅎwnf 1748 ∈ wcel 2030 Ⅎwnfc 2780 ∀wral 2941 Vcvv 3231 ‘cfv 5926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-reg 8538 ax-inf2 8576 ax-ac2 9323 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-en 7998 df-r1 8665 df-rank 8666 df-card 8803 df-ac 8977 |
This theorem is referenced by: (None) |
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