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Mirrors > Home > MPE Home > Th. List > cbvexdva | Structured version Visualization version GIF version |
Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 18-Jul-2021.) |
Ref | Expression |
---|---|
cbvaldva.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvexdva | ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvaldva.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | 1 | notbid 308 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒)) |
3 | 2 | cbvaldva 2281 | . . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒)) |
4 | alnex 1706 | . . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓) | |
5 | alnex 1706 | . . 3 ⊢ (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒) | |
6 | 3, 4, 5 | 3bitr3g 302 | . 2 ⊢ (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒)) |
7 | 6 | con4bid 307 | 1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∃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: cbvex2v 2287 cbvrexdva2 3176 isinf 8173 |
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