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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvrmodavw | Structured version Visualization version GIF version |
Description: Change bound variable in the restricted at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvrmodavw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvrmodavw | ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑦 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2820 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
3 | cbvrmodavw.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | anbi12d 631 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐴 ∧ 𝜒))) |
5 | 4 | cbvmodavw 36193 | . 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜒))) |
6 | df-rmo 3376 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
7 | df-rmo 3376 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜒 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜒)) | |
8 | 5, 6, 7 | 3bitr4g 314 | 1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑦 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2104 ∃*wmo 2534 ∃*wrmo 3375 |
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 ax-8 2106 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-mo 2536 df-clel 2812 df-rmo 3376 |
This theorem is referenced by: cbvdisjdavw 36211 |
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