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Mirrors > Home > MPE Home > Th. List > cbvmo | Structured version Visualization version 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 2260 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
5 | 1, 2, 3 | cbveu 2493 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
6 | 4, 5 | imbi12i 339 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) |
7 | df-mo 2463 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
8 | df-mo 2463 | . 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) | |
9 | 6, 7, 8 | 3bitr4i 291 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∃wex 1695 Ⅎwnf 1699 ∃!weu 2458 ∃*wmo 2459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 |
This theorem is referenced by: dffun6f 5818 opabiotafun 6169 2ndcdisj 21069 cbvdisjf 28767 phpreu 32563 |
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