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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 5715, 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 320 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (¬ 𝜑 ↔ ¬ 𝜓)) |
3 | 2 | raliunxp 5712 | . . . 4 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓) |
4 | ralnex 3238 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
5 | 4 | ralbii 3167 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
6 | 3, 5 | bitri 277 | . . 3 ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
7 | 6 | notbii 322 | . 2 ⊢ (¬ ∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
8 | dfrex2 3241 | . 2 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) ¬ 𝜑) | |
9 | dfrex2 3241 | . 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
10 | 7, 8, 9 | 3bitr4i 305 | 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 208 = wceq 1537 ∀wral 3140 ∃wrex 3141 {csn 4569 〈cop 4575 ∪ ciun 4921 × cxp 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-iun 4923 df-opab 5131 df-xp 5563 df-rel 5564 |
This theorem is referenced by: rexxp 5715 fsumvma 25791 cvmliftlem15 32547 filnetlem4 33731 |
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