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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cpcoll2d | Structured version Visualization version GIF version |
Description: cpcolld 42630 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 5113 | . . . 4 ⊢ (𝑎 = 𝑦 → (𝑥𝐹𝑎 ↔ 𝑥𝐹𝑦)) | |
3 | 2 | cbvexvw 2041 | . . 3 ⊢ (∃𝑎 𝑥𝐹𝑎 ↔ ∃𝑦 𝑥𝐹𝑦) |
4 | 1, 3 | sylibr 233 | . 2 ⊢ (𝜑 → ∃𝑎 𝑥𝐹𝑎) |
5 | cpcoll2d.1 | . . . . 5 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
6 | 5 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → 𝑥 ∈ 𝐴) |
7 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → 𝑥𝐹𝑎) | |
8 | 6, 7 | cpcolld 42630 | . . 3 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → ∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎) |
9 | 2 | cbvrexvw 3225 | . . 3 ⊢ (∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎 ↔ ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) |
10 | 8, 9 | sylib 217 | . 2 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) |
11 | 4, 10 | exlimddv 1939 | 1 ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 ∃wrex 3070 class class class wbr 5109 Coll ccoll 42622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-r1 9708 df-rank 9709 df-scott 42608 df-coll 42623 |
This theorem is referenced by: grumnudlem 42657 |
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