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Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ac6s3f | Structured version Visualization version GIF version |
Description: Generalization of the Axiom of Choice to classes, with bound-variable hypothesis. (Contributed by Giovanni Mascellani, 19-Aug-2018.) |
Ref | Expression |
---|---|
ac6s3f.1 | ⊢ Ⅎ𝑦𝜓 |
ac6s3f.2 | ⊢ 𝐴 ∈ V |
ac6s3f.3 | ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ac6s3f | ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexv 3463 | . . . 4 ⊢ (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑) | |
2 | 1 | ralbii 3132 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦𝜑) |
3 | 2 | biimpri 229 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑) |
4 | ac6s3f.1 | . . 3 ⊢ Ⅎ𝑦𝜓 | |
5 | ac6s3f.2 | . . 3 ⊢ 𝐴 ∈ V | |
6 | ac6s3f.3 | . . 3 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓)) | |
7 | 4, 5, 6 | ac6sf 9762 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 → ∃𝑓(𝑓:𝐴⟶V ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
8 | exsimpr 1851 | . 2 ⊢ (∃𝑓(𝑓:𝐴⟶V ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑓∀𝑥 ∈ 𝐴 𝜓) | |
9 | 3, 7, 8 | 3syl 18 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∃wex 1761 Ⅎwnf 1765 ∈ wcel 2081 ∀wral 3105 ∃wrex 3106 Vcvv 3437 ⟶wf 6226 ‘cfv 6230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-reg 8907 ax-inf2 8955 ax-ac2 9736 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-om 7442 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-en 8363 df-r1 9044 df-rank 9045 df-card 9219 df-ac 9393 |
This theorem is referenced by: ac6s6 35008 |
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