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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvex2vv | Structured version Visualization version GIF version |
Description: Version of cbvex2vv 2427 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbval2vv.1 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-cbvex2vv | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . 2 ⊢ Ⅎ𝑧𝜑 | |
2 | nfv 1906 | . 2 ⊢ Ⅎ𝑤𝜑 | |
3 | nfv 1906 | . 2 ⊢ Ⅎ𝑥𝜓 | |
4 | nfv 1906 | . 2 ⊢ Ⅎ𝑦𝜓 | |
5 | bj-cbval2vv.1 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvex2v 2356 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∃wex 1771 |
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-10 2136 ax-11 2151 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 |
This theorem is referenced by: (None) |
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