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Mirrors > Home > ILE Home > Th. List > cbvmo | GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
cbvmo.1 | ⊢ Ⅎ𝑦𝜑 |
cbvmo.2 | ⊢ Ⅎ𝑥𝜓 |
cbvmo.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvmo | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvmo.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvmo.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvmo.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvex 1639 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
5 | 1, 2, 3 | cbveu 1924 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
6 | 4, 5 | imbi12i 228 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) |
7 | df-mo 1904 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
8 | df-mo 1904 | . 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) | |
9 | 6, 7, 8 | 3bitr4i 201 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 Ⅎwnf 1349 ∃wex 1381 ∃!weu 1900 ∃*wmo 1901 |
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 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 |
This theorem is referenced by: dffun6f 4915 |
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