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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 3182 | . . . . 5 ⊢ ∃𝑧 𝑧 = 𝐴 |
3 | ac6s6f.2 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
4 | vex 3176 | . . . . . 6 ⊢ 𝑧 ∈ V | |
5 | ac6s6f.3 | . . . . . 6 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
6 | 3, 4, 5 | ac6s6 33150 | . . . . 5 ⊢ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓) |
7 | 2, 6 | pm3.2i 470 | . . . 4 ⊢ (∃𝑧 𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
8 | 7 | exan 1775 | . . 3 ⊢ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
9 | exdistr 1906 | . . 3 ⊢ (∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) ↔ ∃𝑧(𝑧 = 𝐴 ∧ ∃𝑓∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓))) | |
10 | 8, 9 | mpbir 220 | . 2 ⊢ ∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) |
11 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
12 | ac6s6f.4 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
13 | 11, 12 | raleqf 3111 | . . . 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 1753 | . 2 ⊢ (∃𝑧∃𝑓(𝑧 = 𝐴 ∧ ∀𝑥 ∈ 𝑧 (∃𝑦𝜑 → 𝜓)) → ∃𝑧∃𝑓∀𝑥 ∈ 𝐴 (∃𝑦𝜑 → 𝜓)) |
16 | ax5e 1829 | . 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 195 ∧ wa 383 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 Vcvv 3173 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 ax-ac2 9168 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-en 7842 df-r1 8510 df-rank 8511 df-card 8648 df-ac 8822 |
This theorem is referenced by: (None) |
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