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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvrex2 | Structured version Visualization version GIF version |
Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3463. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
Ref | Expression |
---|---|
cbvral2.1 | ⊢ Ⅎ𝑧𝜑 |
cbvral2.2 | ⊢ Ⅎ𝑥𝜒 |
cbvral2.3 | ⊢ Ⅎ𝑤𝜒 |
cbvral2.4 | ⊢ Ⅎ𝑦𝜓 |
cbvral2.5 | ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) |
cbvral2.6 | ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrex2 | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑧𝐵 | |
2 | cbvral2.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfrex 3306 | . . 3 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝐵 𝜑 |
4 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | cbvral2.2 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
6 | 4, 5 | nfrex 3306 | . . 3 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐵 𝜒 |
7 | cbvral2.5 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) | |
8 | 7 | rexbidv 3294 | . . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
9 | 3, 6, 8 | cbvrex 3444 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) |
10 | cbvral2.3 | . . . 4 ⊢ Ⅎ𝑤𝜒 | |
11 | cbvral2.4 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
12 | cbvral2.6 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓)) | |
13 | 10, 11, 12 | cbvrex 3444 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓) |
14 | 13 | rexbii 3244 | . 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
15 | 9, 14 | bitri 276 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 Ⅎwnf 1775 ∃wrex 3136 |
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-13 2381 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 |
This theorem is referenced by: (None) |
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