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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnd2d | Structured version Visualization version GIF version |
Description: Deduction form of bnd2 9910. (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 3318 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝐵 𝜓)) | |
4 | raleq 3318 | . . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓 ↔ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓)) | |
5 | 4 | anbi2d 629 | . . . . 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 1917 | . . . 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 344 | . . 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 5301 | . . . . 5 ⊢ ∅ ∈ V | |
9 | 8 | elimel 4593 | . . . 4 ⊢ if(𝐴 ∈ V, 𝐴, ∅) ∈ V |
10 | 9 | bnd2 9910 | . . 3 ⊢ (∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓)) |
11 | 7, 10 | dedth 4582 | . 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 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 Vcvv 3470 ⊆ wss 3945 ∅c0 4318 ifcif 4524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-reg 9609 ax-inf2 9658 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-r1 9781 df-rank 9782 |
This theorem is referenced by: setrec1lem3 48114 |
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