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| Mirrors > Home > MPE Home > Th. List > rexxpf | Structured version Visualization version GIF version | ||
| Description: Version of rexxp 5853 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.) |
| Ref | Expression |
|---|---|
| ralxpf.1 | ⊢ Ⅎ𝑦𝜑 |
| ralxpf.2 | ⊢ Ⅎ𝑧𝜑 |
| ralxpf.3 | ⊢ Ⅎ𝑥𝜓 |
| ralxpf.4 | ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| rexxpf | ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralxpf.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
| 2 | 1 | nfn 1857 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑 |
| 3 | ralxpf.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
| 4 | 3 | nfn 1857 | . . . . 5 ⊢ Ⅎ𝑧 ¬ 𝜑 |
| 5 | ralxpf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
| 6 | 5 | nfn 1857 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝜓 |
| 7 | ralxpf.4 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
| 8 | 7 | notbid 318 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (¬ 𝜑 ↔ ¬ 𝜓)) |
| 9 | 2, 4, 6, 8 | ralxpf 5857 | . . . 4 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓) |
| 10 | ralnex 3072 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
| 11 | 10 | ralbii 3093 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
| 12 | 9, 11 | bitri 275 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
| 13 | 12 | notbii 320 | . 2 ⊢ (¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
| 14 | dfrex2 3073 | . 2 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑) | |
| 15 | dfrex2 3073 | . 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
| 16 | 13, 14, 15 | 3bitr4i 303 | 1 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 Ⅎwnf 1783 ∀wral 3061 ∃wrex 3070 〈cop 4632 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: iunxpf 5859 wdom2d2 43047 |
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