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Mirrors > Home > MPE Home > Th. List > mof | Structured version Visualization version GIF version |
Description: Version of df-mo 2535 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2591 from this proof, and prove mof 2558 from it (as of 30-Sep-2022, directly from df-mo 2535). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2372. (Revised by Wolf Lammen, 16-Oct-2022.) |
Ref | Expression |
---|---|
mof.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
mof | ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2535 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
2 | mof.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
4 | 2, 3 | nfim 1900 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧) |
5 | 4 | nfal 2317 | . . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑧) |
6 | nfv 1918 | . . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 → 𝑥 = 𝑦) | |
7 | equequ2 2030 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
8 | 7 | imbi2d 341 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜑 → 𝑥 = 𝑦))) |
9 | 8 | albidv 1924 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))) |
10 | 5, 6, 9 | cbvexv1 2339 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
11 | 1, 10 | bitri 275 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∃wex 1782 Ⅎwnf 1786 ∃*wmo 2533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 df-mo 2535 |
This theorem is referenced by: mo3 2559 mo 2560 rmo2 3882 nmo 31730 fineqvrep 34095 bj-eu3f 35720 dffun3f 47727 |
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