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Mirrors > Home > MPE Home > Th. List > rexxpf | Structured version Visualization version GIF version |
Description: Version of rexxp 5706 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 1848 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑 |
3 | ralxpf.2 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
4 | 3 | nfn 1848 | . . . . 5 ⊢ Ⅎ𝑧 ¬ 𝜑 |
5 | ralxpf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | nfn 1848 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝜓 |
7 | ralxpf.4 | . . . . . 6 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝜑 ↔ 𝜓)) | |
8 | 7 | notbid 319 | . . . . 5 ⊢ (𝑥 = 〈𝑦, 𝑧〉 → (¬ 𝜑 ↔ ¬ 𝜓)) |
9 | 2, 4, 6, 8 | ralxpf 5710 | . . . 4 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓) |
10 | ralnex 3233 | . . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
11 | 10 | ralbii 3162 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
12 | 9, 11 | bitri 276 | . . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
13 | 12 | notbii 321 | . 2 ⊢ (¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) |
14 | dfrex2 3236 | . 2 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑) | |
15 | dfrex2 3236 | . 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓) | |
16 | 13, 14, 15 | 3bitr4i 304 | 1 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1528 Ⅎwnf 1775 ∀wral 3135 ∃wrex 3136 〈cop 4563 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-iun 4912 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: iunxpf 5712 wdom2d2 39510 |
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