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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 2418, 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 37752 is sb5ALTVD 38171 without virtual deductions and
was automatically derived from sb5ALTVD 38171.
|
Ref | Expression |
---|---|
sb5ALTVD | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn1 37811 | . . . . . 6 ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥]𝜑 ) | |
2 | equsb1 2356 | . . . . . 6 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | |
3 | sban 2387 | . . . . . . 7 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑)) | |
4 | 3 | simplbi2com 655 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑))) |
5 | 1, 2, 4 | e10 37940 | . . . . 5 ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) ) |
6 | spsbe 1871 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
7 | 5, 6 | e1a 37873 | . . . 4 ⊢ ( [𝑦 / 𝑥]𝜑 ▶ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ) |
8 | 7 | in1 37808 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
9 | hbs1 2424 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | |
10 | idn2 37859 | . . . . . 6 ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑) ▶ (𝑥 = 𝑦 ∧ 𝜑) ) | |
11 | simpr 476 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜑) | |
12 | 10, 11 | e2 37877 | . . . . 5 ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑) ▶ 𝜑 ) |
13 | simpl 472 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝑥 = 𝑦) | |
14 | 10, 13 | e2 37877 | . . . . 5 ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑) ▶ 𝑥 = 𝑦 ) |
15 | sbequ1 2096 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
16 | 15 | com12 32 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑)) |
17 | 12, 14, 16 | e22 37917 | . . . 4 ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑) ▶ [𝑦 / 𝑥]𝜑 ) |
18 | 9, 17 | exinst 37870 | . . 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 197 | . . 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 444 | . 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 38020 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-vd1 37807 df-vd2 37815 |
This theorem is referenced by: (None) |
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