| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > rmobidv | Structured version Visualization version GIF version | ||
| Description: Formula-building rule for restricted at-most-one quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) |
| Ref | Expression |
|---|---|
| rmobidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| rmobidv | ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rmobidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
| 3 | 2 | rmobidva 3395 | 1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 ∃*wrmo 3379 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-mo 2540 df-rmo 3380 |
| This theorem is referenced by: rmoeqd 3420 brdom7disj 10571 ddemeas 34237 poimirlem26 37653 |
| Copyright terms: Public domain | W3C validator |