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Mathbox for Alexander van der Vekens |
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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 3375. (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 2933 | . . . 4 ⊢ Ⅎ𝑧𝐵 | |
2 | cbvral2.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
3 | 1, 2 | nfrex 3187 | . . 3 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝐵 𝜑 |
4 | nfcv 2933 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | cbvral2.2 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
6 | 4, 5 | nfrex 3187 | . . 3 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐵 𝜒 |
7 | cbvral2.5 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒)) | |
8 | 7 | rexbidv 3236 | . . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
9 | 3, 6, 8 | cbvrex 3363 | . 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 3363 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓) |
14 | 13 | rexbii 3223 | . 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
15 | 9, 14 | bitri 267 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 Ⅎwnf 1827 ∃wrex 3090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 |
This theorem is referenced by: (None) |
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