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Mirrors > Home > MPE Home > Th. List > mof | Structured version Visualization version GIF version |
Description: Version of df-mo 2539 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2595 from this proof, and prove mof 2562 from it (as of 30-Sep-2022, directly from df-mo 2539). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2371. (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 2539 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
2 | mof.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1922 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
4 | 2, 3 | nfim 1904 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧) |
5 | 4 | nfal 2324 | . . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑧) |
6 | nfv 1922 | . . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 → 𝑥 = 𝑦) | |
7 | equequ2 2036 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
8 | 7 | imbi2d 344 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜑 → 𝑥 = 𝑦))) |
9 | 8 | albidv 1928 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))) |
10 | 5, 6, 9 | cbvexv1 2343 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
11 | 1, 10 | bitri 278 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 ∃wex 1787 Ⅎwnf 1791 ∃*wmo 2537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-10 2143 ax-11 2160 ax-12 2177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-mo 2539 |
This theorem is referenced by: mo3 2563 mo 2564 rmo2 3786 nmo 30511 fineqvrep 32731 bj-eu3f 34711 dffun3f 46002 |
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