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Mirrors > Home > MPE Home > Th. List > rexxpf | Structured version Visualization version GIF version |
Description: Version of rexxp 5849 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 1852 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑 |
3 | ralxpf.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
4 | 3 | nfn 1852 | . . . . 5 ⊢ Ⅎ𝑧 ¬ 𝜑 |
5 | ralxpf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1852 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝜓 |
7 | ralxpf.4 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
8 | 7 | notbid 317 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (¬ 𝜑 ↔ ¬ 𝜓)) |
9 | 2, 4, 6, 8 | ralxpf 5853 | . . . 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 274 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
13 | 12 | notbii 319 | . 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 302 | 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 1533 Ⅎwnf 1777 ∀wral 3058 ∃wrex 3067 〈cop 4638 × cxp 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-iun 5002 df-opab 5215 df-xp 5688 df-rel 5689 |
This theorem is referenced by: iunxpf 5855 wdom2d2 42487 |
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