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Mirrors > Home > MPE Home > Th. List > mojust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-mo 2598 (note that 𝑦 and 𝑧 need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). (Contributed by NM, 11-Mar-2010.) Added this theorem by adapting the proof of eujust 2631. (Revised by BJ, 30-Sep-2022.) |
Ref | Expression |
---|---|
mojust | ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equequ2 2033 | . . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡)) | |
2 | 1 | imbi2d 344 | . . . 4 ⊢ (𝑦 = 𝑡 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝑡))) |
3 | 2 | albidv 1921 | . . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑡))) |
4 | 3 | cbvexvw 2044 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑡∀𝑥(𝜑 → 𝑥 = 𝑡)) |
5 | equequ2 2033 | . . . . 5 ⊢ (𝑡 = 𝑧 → (𝑥 = 𝑡 ↔ 𝑥 = 𝑧)) | |
6 | 5 | imbi2d 344 | . . . 4 ⊢ (𝑡 = 𝑧 → ((𝜑 → 𝑥 = 𝑡) ↔ (𝜑 → 𝑥 = 𝑧))) |
7 | 6 | albidv 1921 | . . 3 ⊢ (𝑡 = 𝑧 → (∀𝑥(𝜑 → 𝑥 = 𝑡) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑧))) |
8 | 7 | cbvexvw 2044 | . 2 ⊢ (∃𝑡∀𝑥(𝜑 → 𝑥 = 𝑡) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) |
9 | 4, 8 | bitri 278 | 1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: (None) |
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