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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rmoeqbidv | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for restricted at-most-one quantifier. Deduction form. General version of rmobidv 3396. (Contributed by GG, 1-Sep-2025.) |
| Ref | Expression |
|---|---|
| rmoeqbidv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| rmoeqbidv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| rmoeqbidv | ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rmoeqbidv.1 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 1 | eleq2d 2826 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 3 | rmoeqbidv.2 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 4 | 2, 3 | anbi12d 632 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
| 5 | 4 | mobidv 2548 | . 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))) |
| 6 | df-rmo 3379 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
| 7 | df-rmo 3379 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜒 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)) | |
| 8 | 5, 6, 7 | 3bitr4g 314 | 1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃*wmo 2537 ∃*wrmo 3378 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-mo 2539 df-cleq 2728 df-clel 2815 df-rmo 3379 |
| This theorem is referenced by: (None) |
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