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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 2859 | . . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃!𝑦 𝑦 = 𝐴) | |
2 | isset 2695 | . . . . . 6 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴) | |
3 | 1, 2 | bitr3i 185 | . . . . 5 ⊢ (∃!𝑦 𝑦 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴) |
4 | nfv 1509 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴 | |
5 | 4 | euexex 2085 | . . . . 5 ⊢ ((∃!𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) |
6 | 3, 5 | sylanbr 283 | . . . 4 ⊢ ((∃𝑦 𝑦 = 𝐴 ∧ ∀𝑦∃*𝑥𝜑) → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) |
7 | 6 | expcom 115 | . . 3 ⊢ (∀𝑦∃*𝑥𝜑 → (∃𝑦 𝑦 = 𝐴 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))) |
8 | moanimv 2075 | . . 3 ⊢ (∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) ↔ (∃𝑦 𝑦 = 𝐴 → ∃*𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝜑))) | |
9 | 7, 8 | sylibr 133 | . 2 ⊢ (∀𝑦∃*𝑥𝜑 → ∃*𝑥(∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))) |
10 | simpl 108 | . . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝜑) → 𝑦 = 𝐴) | |
11 | 10 | eximi 1580 | . . . 4 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → ∃𝑦 𝑦 = 𝐴) |
12 | 11 | ancri 322 | . . 3 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) → (∃𝑦 𝑦 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))) |
13 | 12 | moimi 2065 | . 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 103 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∃!weu 2000 ∃*wmo 2001 Vcvv 2689 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: mosub 2866 |
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