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Mirrors > Home > MPE Home > Th. List > cbvex2vw | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2vv 2417 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995.) Avoid ax-13 2375. (Revised by GG, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbval2vw.1 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvex2vw | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval2vw.1 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
2 | 1 | cbvexdvaw 2036 | . 2 ⊢ (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓)) |
3 | 2 | cbvexvw 2034 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: cbvex4vw 2039 cbvopabv 5221 cbvoprab12v 7523 cbvoprab123vw 36222 cbvoprab23vw 36223 bj-cbvex4vv 36788 funop1 47233 uspgrsprf1 47991 |
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