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Mirrors > Home > MPE Home > Th. List > rexxpf | Structured version Visualization version GIF version |
Description: Version of rexxp 5845 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 1853 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑 |
3 | ralxpf.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
4 | 3 | nfn 1853 | . . . . 5 ⊢ Ⅎ𝑧 ¬ 𝜑 |
5 | ralxpf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1853 | . . . . 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 5849 | . . . 4 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓) |
10 | ralnex 3069 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
11 | 10 | ralbii 3090 | . . . 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 3070 | . 2 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑) | |
15 | dfrex2 3070 | . 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 205 = wceq 1534 Ⅎwnf 1778 ∀wral 3058 ∃wrex 3067 ⟨cop 4635 × cxp 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-iun 4998 df-opab 5211 df-xp 5684 df-rel 5685 |
This theorem is referenced by: iunxpf 5851 wdom2d2 42456 |
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