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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sb5ALTVD | Structured version Visualization version GIF version | ||
| Description: The following User's Proof is a Natural Deduction Sequent Calculus
       transcription of the Fitch-style Natural Deduction proof of Unit 20
       Excercise 3.a., which is sb5 2275, found in the "Answers to Starred
       Exercises" on page 457 of "Understanding Symbolic Logic", Fifth
       Edition (2008), by Virginia Klenk.  The same proof may also be
       interpreted as a Virtual Deduction Hilbert-style axiomatic proof.  It
       was completed automatically by the tools program completeusersproof.cmd,
       which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof
       Assistant.  sb5ALT 44550 is sb5ALTVD 44938 without virtual deductions and
       was automatically derived from sb5ALTVD 44938. 
 | 
| Ref | Expression | 
|---|---|
| sb5ALTVD | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | idn1 44599 | . . . . . 6 ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥]𝜑 ) | |
| 2 | equsb1 2495 | . . . . . 6 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | |
| 3 | sban 2079 | . . . . . . 7 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑)) | |
| 4 | 3 | simplbi2com 502 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑))) | 
| 5 | 1, 2, 4 | e10 44719 | . . . . 5 ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) ) | 
| 6 | spsbe 2081 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
| 7 | 5, 6 | e1a 44652 | . . . 4 ⊢ ( [𝑦 / 𝑥]𝜑 ▶ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ) | 
| 8 | 7 | in1 44596 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | 
| 9 | hbs1 2273 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
| 10 | idn2 44638 | . . . . . 6 ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑) ▶ (𝑥 = 𝑦 ∧ 𝜑) ) | |
| 11 | simpr 484 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑) | |
| 12 | 10, 11 | e2 44656 | . . . . 5 ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑) ▶ 𝜑 ) | 
| 13 | simpl 482 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦) | |
| 14 | 10, 13 | e2 44656 | . . . . 5 ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑) ▶ 𝑥 = 𝑦 ) | 
| 15 | sbequ1 2247 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
| 16 | 15 | com12 32 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑)) | 
| 17 | 12, 14, 16 | e22 44696 | . . . 4 ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑) ▶ [𝑦 / 𝑥]𝜑 ) | 
| 18 | 9, 17 | exinst 44649 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑) | 
| 19 | 8, 18 | pm3.2i 470 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ∧ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)) | 
| 20 | impbi 208 | . . 3 ⊢ (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ((∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))) | |
| 21 | 20 | imp 406 | . 2 ⊢ ((([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ∧ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | 
| 22 | 19, 21 | e0a 44797 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 [wsb 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-12 2176 ax-13 2376 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1779 df-nf 1783 df-sb 2064 df-vd1 44595 df-vd2 44603 | 
| This theorem is referenced by: (None) | 
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