Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cbvex4vw | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvex4v 2437 with more disjoint variable conditions, which requires fewer axioms. (Contributed by NM, 26-Jul-1995.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvex4vw.1 | ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) |
cbvex4vw.2 | ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvex4vw | ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvex4vw.1 | . . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) | |
2 | 1 | 2exbidv 1925 | . . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤𝜓)) |
3 | 2 | cbvex2vw 2048 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑧∃𝑤𝜓) |
4 | cbvex4vw.2 | . . . 4 ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) | |
5 | 4 | cbvex2vw 2048 | . . 3 ⊢ (∃𝑧∃𝑤𝜓 ↔ ∃𝑓∃𝑔𝜒) |
6 | 5 | 2exbii 1849 | . 2 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤𝜓 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
7 | 3, 6 | bitri 277 | 1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: addsrmo 10495 mulsrmo 10496 |
Copyright terms: Public domain | W3C validator |