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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 2530 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
5 | 1, 2, 3 | cbvreu 2531 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓) |
6 | 4, 5 | imbi12i 228 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) |
7 | rmo5 2525 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | |
8 | rmo5 2525 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 → ∃!𝑦 ∈ 𝐴 𝜓)) | |
9 | 6, 7, 8 | 3bitr4i 201 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 Ⅎwnf 1349 ∃wrex 2307 ∃!wreu 2308 ∃*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 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-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-reu 2313 df-rmo 2314 |
This theorem is referenced by: cbvrmov 2536 cbvdisj 3755 |
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