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Mirrors > Home > MPE Home > Th. List > cbvex2v | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 18-Jul-2021.) |
Ref | Expression |
---|---|
cbval2v.1 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvex2v | ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval2v.1 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
2 | 1 | cbvexdva 2283 | . 2 ⊢ (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓)) |
3 | 2 | cbvexv 2275 | 1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-11 2034 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: cbvex4v 2289 funop1 41302 uspgrsprf1 41755 |
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