Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mojust | Structured version Visualization version GIF version |
Description: Soundness justification theorem for df-mo 2616 (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 2650. (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 2027 | . . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡)) | |
2 | 1 | imbi2d 343 | . . . 4 ⊢ (𝑦 = 𝑡 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝑡))) |
3 | 2 | albidv 1915 | . . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑡))) |
4 | 3 | cbvexvw 2038 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑡∀𝑥(𝜑 → 𝑥 = 𝑡)) |
5 | equequ2 2027 | . . . . 5 ⊢ (𝑡 = 𝑧 → (𝑥 = 𝑡 ↔ 𝑥 = 𝑧)) | |
6 | 5 | imbi2d 343 | . . . 4 ⊢ (𝑡 = 𝑧 → ((𝜑 → 𝑥 = 𝑡) ↔ (𝜑 → 𝑥 = 𝑧))) |
7 | 6 | albidv 1915 | . . 3 ⊢ (𝑡 = 𝑧 → (∀𝑥(𝜑 → 𝑥 = 𝑡) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑧))) |
8 | 7 | cbvexvw 2038 | . 2 ⊢ (∃𝑡∀𝑥(𝜑 → 𝑥 = 𝑡) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) |
9 | 4, 8 | bitri 277 | 1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1529 ∃wex 1774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1775 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |