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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 3490 | . . . 4 ⊢ ∃𝑧 𝑧 = 𝐴 |
3 | ac6s6f.2 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
4 | vex 3479 | . . . . 5 ⊢ 𝑧 ∈ V | |
5 | ac6s6f.3 | . . . . 5 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
6 | 3, 4, 5 | ac6s6 37088 | . . . 4 ⊢ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) |
7 | 2, 6 | exan 1866 | . . 3 ⊢ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
8 | exdistr 1959 | . . 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 3350 | . . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓))) |
13 | 12 | biimpa 478 | . . 3 ⊢ ((𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) → ∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) |
14 | 13 | 2eximi 1839 | . 2 ⊢ (∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) → ∃𝑧∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) |
15 | ax5e 1916 | . 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 397 = wceq 1542 ∃wex 1782 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ∀wral 3062 Vcvv 3475 ‘cfv 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-reg 9587 ax-inf2 9636 ax-ac2 10458 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-en 8940 df-r1 9759 df-rank 9760 df-card 9934 df-ac 10111 |
This theorem is referenced by: (None) |
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