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| Description: Deduction form of bnd2 9933. (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 3323 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝐵 𝜓)) | |
| 4 | raleq 3323 | . . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ V, 𝐴, ∅) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜓 ↔ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓)) | |
| 5 | 4 | anbi2d 630 | . . . . 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 1921 | . . . 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 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 9 | 8 | elimel 4595 | . . . 4 ⊢ if(𝐴 ∈ V, 𝐴, ∅) ∈ V | 
| 10 | 9 | bnd2 9933 | . . 3 ⊢ (∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ if (𝐴 ∈ V, 𝐴, ∅)∃𝑦 ∈ 𝑧 𝜓)) | 
| 11 | 7, 10 | dedth 4584 | . 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 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 ifcif 4525 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-r1 9804 df-rank 9805 | 
| This theorem is referenced by: setrec1lem3 49208 | 
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