![]() |
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 3392 | 1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2105 ∃*wrmo 3376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-mo 2537 df-rmo 3377 |
This theorem is referenced by: rmoeqd 3417 brdom7disj 10568 ddemeas 34216 poimirlem26 37632 |
Copyright terms: Public domain | W3C validator |