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Mirrors > Home > ILE Home > Th. List > cbvex4v | GIF version |
Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 26-Jul-1995.) |
Ref | Expression |
---|---|
cbvex4v.1 | ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) |
cbvex4v.2 | ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvex4v | ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvex4v.1 | . . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) | |
2 | 1 | 2exbidv 1861 | . . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤𝜓)) |
3 | 2 | cbvex2v 1917 | . 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑧∃𝑤𝜓) |
4 | cbvex4v.2 | . . . 4 ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) | |
5 | 4 | cbvex2v 1917 | . . 3 ⊢ (∃𝑧∃𝑤𝜓 ↔ ∃𝑓∃𝑔𝜒) |
6 | 5 | 2exbii 1599 | . 2 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤𝜓 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
7 | 3, 6 | bitri 183 | 1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∃wex 1485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: enq0sym 7394 addnq0mo 7409 mulnq0mo 7410 addsrmo 7705 mulsrmo 7706 |
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