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Mirrors > Home > MPE Home > Th. List > reueqdv | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 1-Sep-2025.) |
Ref | Expression |
---|---|
reueqdv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
reueqdv | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reueqdv.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | reueq1 3414 | . 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1535 ∃!wreu 3374 |
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-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1775 df-mo 2536 df-eu 2565 df-cleq 2725 df-rex 3067 df-rmo 3376 df-reu 3377 |
This theorem is referenced by: upciclem1 48733 |
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