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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cpcoll2d | Structured version Visualization version GIF version | ||
| Description: cpcolld 44282 with an extra existential quantifier. (Contributed by Rohan Ridenour, 12-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| cpcoll2d.1 | ⊢ (𝜑 → 𝑥 ∈ 𝐴) | 
| cpcoll2d.2 | ⊢ (𝜑 → ∃𝑦 𝑥𝐹𝑦) | 
| Ref | Expression | 
|---|---|
| cpcoll2d | ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cpcoll2d.2 | . . 3 ⊢ (𝜑 → ∃𝑦 𝑥𝐹𝑦) | |
| 2 | breq2 5146 | . . . 4 ⊢ (𝑎 = 𝑦 → (𝑥𝐹𝑎 ↔ 𝑥𝐹𝑦)) | |
| 3 | 2 | cbvexvw 2035 | . . 3 ⊢ (∃𝑎 𝑥𝐹𝑎 ↔ ∃𝑦 𝑥𝐹𝑦) | 
| 4 | 1, 3 | sylibr 234 | . 2 ⊢ (𝜑 → ∃𝑎 𝑥𝐹𝑎) | 
| 5 | cpcoll2d.1 | . . . . 5 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
| 6 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → 𝑥 ∈ 𝐴) | 
| 7 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → 𝑥𝐹𝑎) | |
| 8 | 6, 7 | cpcolld 44282 | . . 3 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → ∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎) | 
| 9 | 2 | cbvrexvw 3237 | . . 3 ⊢ (∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎 ↔ ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) | 
| 10 | 8, 9 | sylib 218 | . 2 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) | 
| 11 | 4, 10 | exlimddv 1934 | 1 ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1778 ∈ wcel 2107 ∃wrex 3069 class class class wbr 5142 Coll ccoll 44274 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-r1 9805 df-rank 9806 df-scott 44260 df-coll 44275 | 
| This theorem is referenced by: grumnudlem 44309 | 
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