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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvmodavw | Structured version Visualization version GIF version |
Description: Change bound variable in the at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvmodavw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvmodavw | ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvmodavw.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | equequ1 2020 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
3 | 2 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
4 | 1, 3 | imbi12d 344 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝜓 → 𝑥 = 𝑧) ↔ (𝜒 → 𝑦 = 𝑧))) |
5 | 4 | cbvaldvaw 2033 | . . 3 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜒 → 𝑦 = 𝑧))) |
6 | 5 | exbidv 1917 | . 2 ⊢ (𝜑 → (∃𝑧∀𝑥(𝜓 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜒 → 𝑦 = 𝑧))) |
7 | df-mo 2536 | . 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑧∀𝑥(𝜓 → 𝑥 = 𝑧)) | |
8 | df-mo 2536 | . 2 ⊢ (∃*𝑦𝜒 ↔ ∃𝑧∀𝑦(𝜒 → 𝑦 = 𝑧)) | |
9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑦𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1533 ∃wex 1774 ∃*wmo 2534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-mo 2536 |
This theorem is referenced by: cbveudavw 36194 cbvrmodavw 36195 cbvrmodavw2 36226 |
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