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Mirrors > Home > MPE Home > Th. List > rexiunxp | Structured version Visualization version GIF version |
Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp 5696, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.) |
Ref | Expression |
---|---|
ralxp.1 | ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexiunxp | ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralxp.1 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
2 | 1 | notbid 321 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | 2 | raliunxp 5693 | . . . 4 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓) |
4 | ralnex 3148 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
5 | 4 | ralbii 3078 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
6 | 3, 5 | bitri 278 | . . 3 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
7 | 6 | notbii 323 | . 2 ⊢ (¬ ∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
8 | dfrex2 3151 | . 2 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ¬ 𝜑) | |
9 | dfrex2 3151 | . 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
10 | 7, 8, 9 | 3bitr4i 306 | 1 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1543 ∀wral 3051 ∃wrex 3052 {csn 4527 〈cop 4533 ∪ ciun 4890 × cxp 5534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-iun 4892 df-opab 5102 df-xp 5542 df-rel 5543 |
This theorem is referenced by: rexxp 5696 fsumvma 26048 cvmliftlem15 32927 filnetlem4 34256 |
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