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Mirrors > Home > ILE Home > Th. List > rmoan | GIF version |
Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
rmoan | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moan 2066 | . . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | an12 550 | . . . 4 ⊢ ((𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑))) | |
3 | 2 | mobii 2034 | . . 3 ⊢ (∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑))) |
4 | 1, 3 | sylib 121 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑))) |
5 | df-rmo 2422 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | df-rmo 2422 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑))) | |
7 | 4, 5, 6 | 3imtr4i 200 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ∃*wmo 1998 ∃*wrmo 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-rmo 2422 |
This theorem is referenced by: (None) |
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