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Mirrors > Home > ILE Home > Th. List > cbvrmo | GIF version |
Description: Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
cbvral.1 | ⊢ Ⅎ𝑦𝜑 |
cbvral.2 | ⊢ Ⅎ𝑥𝜓 |
cbvral.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrmo | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvral.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvral.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvral.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvrex 2649 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
5 | 1, 2, 3 | cbvreu 2650 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
6 | 4, 5 | imbi12i 238 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) |
7 | rmo5 2644 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
8 | rmo5 2644 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) | |
9 | 6, 7, 8 | 3bitr4i 211 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 Ⅎwnf 1436 ∃wrex 2415 ∃!wreu 2416 ∃*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 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-reu 2421 df-rmo 2422 |
This theorem is referenced by: cbvrmov 2655 cbvdisj 3911 |
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