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Mirrors > Home > MPE Home > Th. List > mof | Structured version Visualization version GIF version |
Description: Version of df-mo 2538 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2594 from this proof, and prove mof 2561 from it (as of 30-Sep-2022, directly from df-mo 2538). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2370. (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 2538 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
2 | mof.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
3 | nfv 1916 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
4 | 2, 3 | nfim 1898 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧) |
5 | 4 | nfal 2316 | . . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑧) |
6 | nfv 1916 | . . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 → 𝑥 = 𝑦) | |
7 | equequ2 2028 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦)) | |
8 | 7 | imbi2d 340 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜑 → 𝑥 = 𝑦))) |
9 | 8 | albidv 1922 | . . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))) |
10 | 5, 6, 9 | cbvexv1 2338 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
11 | 1, 10 | bitri 274 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1538 ∃wex 1780 Ⅎwnf 1784 ∃*wmo 2536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-10 2136 ax-11 2153 ax-12 2170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1781 df-nf 1785 df-mo 2538 |
This theorem is referenced by: mo3 2562 mo 2563 rmo2 3831 nmo 31126 fineqvrep 33363 bj-eu3f 35120 dffun3f 46748 |
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