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Mirrors > Home > ILE Home > Th. List > rmoim | GIF version |
Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Ref | Expression |
---|---|
rmoim | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2311 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
2 | imdistan 418 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) | |
3 | 2 | albii 1359 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
4 | 1, 3 | bitri 173 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
5 | moim 1964 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
6 | df-rmo 2314 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
7 | df-rmo 2314 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
8 | 5, 6, 7 | 3imtr4g 194 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) |
9 | 4, 8 | sylbi 114 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 ∈ wcel 1393 ∃*wmo 1901 ∀wral 2306 ∃*wrmo 2309 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-ral 2311 df-rmo 2314 |
This theorem is referenced by: rmoimia 2741 disjss2 3748 |
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