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Mirrors > Home > ILE Home > Th. List > mosubt | GIF version |
Description: "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.) |
Ref | Expression |
---|---|
mosubt | ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eueq 2712 | . . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃!𝑦 𝑦 = 𝐴) | |
2 | isset 2561 | . . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
3 | 1, 2 | bitr3i 175 | . . . . 5 ⊢ (∃!𝑦 𝑦 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
4 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴 | |
5 | 4 | euexex 1985 | . . . . 5 ⊢ ((∃!𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) |
6 | 3, 5 | sylanbr 269 | . . . 4 ⊢ ((∃𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) |
7 | 6 | expcom 109 | . . 3 ⊢ (∀𝑦∃*𝑥𝜑 → (∃𝑦 𝑦 = 𝐴 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))) |
8 | moanimv 1975 | . . 3 ⊢ (∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) ↔ (∃𝑦 𝑦 = 𝐴 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))) | |
9 | 7, 8 | sylibr 137 | . 2 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))) |
10 | simpl 102 | . . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝜑) → 𝑦 = 𝐴) | |
11 | 10 | eximi 1491 | . . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → ∃𝑦 𝑦 = 𝐴) |
12 | 11 | ancri 307 | . . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → (∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))) |
13 | 12 | moimi 1965 | . 2 ⊢ (∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) |
14 | 9, 13 | syl 14 | 1 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃!weu 1900 ∃*wmo 1901 Vcvv 2557 |
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 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: mosub 2719 |
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