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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnd2d | Structured version Visualization version GIF version |
Description: Deduction form of bnd2 9055. (Contributed by Emmett Weisz, 19-Jan-2021.) |
Ref | Expression |
---|---|
bnd2d.1 | ⊢ (𝜑 → 𝐴 ∈ V) |
bnd2d.2 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) |
Ref | Expression |
---|---|
bnd2d | ⊢ (𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnd2d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) | |
2 | bnd2d.2 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) | |
3 | raleq 3330 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝐵 𝜓)) | |
4 | raleq 3330 | . . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓 ↔ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓)) | |
5 | 4 | anbi2d 622 | . . . . 5 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → ((𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓) ↔ (𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓))) |
6 | 5 | exbidv 1964 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → (∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓) ↔ ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓))) |
7 | 3, 6 | imbi12d 336 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → ((∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓)) ↔ (∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓)))) |
8 | 0ex 5028 | . . . . 5 ⊢ ∅ ∈ V | |
9 | 8 | elimel 4374 | . . . 4 ⊢ if(𝐴 ∈ V, 𝐴, ∅) ∈ V |
10 | 9 | bnd2 9055 | . . 3 ⊢ (∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓)) |
11 | 7, 10 | dedth 4363 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓))) |
12 | 1, 2, 11 | sylc 65 | 1 ⊢ (𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ⊆ wss 3792 ∅c0 4141 ifcif 4307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-reg 8788 ax-inf2 8837 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-r1 8926 df-rank 8927 |
This theorem is referenced by: setrec1lem3 43555 |
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