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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 44254 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 5152 | . . . 4 ⊢ (𝑎 = 𝑦 → (𝑥𝐹𝑎 ↔ 𝑥𝐹𝑦)) | |
3 | 2 | cbvexvw 2034 | . . 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 44254 | . . 3 ⊢ ((𝜑 ∧ 𝑥𝐹𝑎) → ∃𝑎 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑎) |
9 | 2 | cbvrexvw 3236 | . . 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 1933 | 1 ⊢ (𝜑 → ∃𝑦 ∈ (𝐹 Coll 𝐴)𝑥𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 Coll ccoll 44246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-r1 9802 df-rank 9803 df-scott 44232 df-coll 44247 |
This theorem is referenced by: grumnudlem 44281 |
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