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Theorem sb5ALTVD 44938
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.
1:: (   [𝑦 / 𝑥]𝜑   ▶   [𝑦 / 𝑥]𝜑   )
2:: [𝑦 / 𝑥]𝑥 = 𝑦
3:1,2: (   [𝑦 / 𝑥]𝜑   ▶   [𝑦 / 𝑥](𝑥 = 𝑦 𝜑)   )
4:3: (   [𝑦 / 𝑥]𝜑   ▶   𝑥(𝑥 = 𝑦𝜑 )   )
5:4: ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑) )
6:: (   𝑥(𝑥 = 𝑦𝜑)   ▶   𝑥(𝑥 = 𝑦𝜑)   )
7:: (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑 )   ▶   (𝑥 = 𝑦𝜑)   )
8:7: (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑 )   ▶   𝜑   )
9:7: (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑 )   ▶   𝑥 = 𝑦   )
10:8,9: (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑 )   ▶   [𝑦 / 𝑥]𝜑   )
101:: ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
11:101,10: (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑 )
12:5,11: (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑 )) ∧ (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
qed:12: ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑) )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb5ALTVD ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb5ALTVD
StepHypRef 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 = 𝑦𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑))
43simplbi2com 502 . . . . . 6 ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝑥 = 𝑦𝜑)))
51, 2, 4e10 44719 . . . . 5 (   [𝑦 / 𝑥]𝜑   ▶   [𝑦 / 𝑥](𝑥 = 𝑦𝜑)   )
6 spsbe 2081 . . . . 5 ([𝑦 / 𝑥](𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
75, 6e1a 44652 . . . 4 (   [𝑦 / 𝑥]𝜑   ▶   𝑥(𝑥 = 𝑦𝜑)   )
87in1 44596 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
9 hbs1 2273 . . . 4 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
10 idn2 44638 . . . . . 6 (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑)   ▶   (𝑥 = 𝑦𝜑)   )
11 simpr 484 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝜑)
1210, 11e2 44656 . . . . 5 (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑)   ▶   𝜑   )
13 simpl 482 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝑥 = 𝑦)
1410, 13e2 44656 . . . . 5 (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑)   ▶   𝑥 = 𝑦   )
15 sbequ1 2247 . . . . . 6 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
1615com12 32 . . . . 5 (𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑))
1712, 14, 16e22 44696 . . . 4 (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑)   ▶   [𝑦 / 𝑥]𝜑   )
189, 17exinst 44649 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
198, 18pm3.2i 470 . 2 (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)) ∧ (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
20 impbi 208 . . 3 (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)) → ((∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))))
2120imp 406 . 2 ((([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)) ∧ (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
2219, 21e0a 44797 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
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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