|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 2418 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995.) Avoid ax-13 2376. (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 2037 | . 2 ⊢ (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓)) | 
| 3 | 2 | cbvexvw 2035 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1778 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: cbvex4vw 2040 cbvopabv 5215 cbvoprab12v 7524 cbvoprab123vw 36241 cbvoprab23vw 36242 bj-cbvex4vv 36807 funop1 47300 uspgrsprf1 48068 | 
| Copyright terms: Public domain | W3C validator |