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Theorem sb5ALTVD 38969
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 2428, 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 38551 is sb5ALTVD 38969 without virtual deductions and was automatically derived from sb5ALTVD 38969.
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 38610 . . . . . 6 (   [𝑦 / 𝑥]𝜑   ▶   [𝑦 / 𝑥]𝜑   )
2 equsb1 2366 . . . . . 6 [𝑦 / 𝑥]𝑥 = 𝑦
3 sban 2397 . . . . . . 7 ([𝑦 / 𝑥](𝑥 = 𝑦𝜑) ↔ ([𝑦 / 𝑥]𝑥 = 𝑦 ∧ [𝑦 / 𝑥]𝜑))
43simplbi2com 656 . . . . . 6 ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝑥 = 𝑦 → [𝑦 / 𝑥](𝑥 = 𝑦𝜑)))
51, 2, 4e10 38739 . . . . 5 (   [𝑦 / 𝑥]𝜑   ▶   [𝑦 / 𝑥](𝑥 = 𝑦𝜑)   )
6 spsbe 1882 . . . . 5 ([𝑦 / 𝑥](𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
75, 6e1a 38672 . . . 4 (   [𝑦 / 𝑥]𝜑   ▶   𝑥(𝑥 = 𝑦𝜑)   )
87in1 38607 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
9 hbs1 2434 . . . 4 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
10 idn2 38658 . . . . . 6 (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑)   ▶   (𝑥 = 𝑦𝜑)   )
11 simpr 477 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝜑)
1210, 11e2 38676 . . . . 5 (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑)   ▶   𝜑   )
13 simpl 473 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝑥 = 𝑦)
1410, 13e2 38676 . . . . 5 (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑)   ▶   𝑥 = 𝑦   )
15 sbequ1 2108 . . . . . 6 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
1615com12 32 . . . . 5 (𝜑 → (𝑥 = 𝑦 → [𝑦 / 𝑥]𝜑))
1712, 14, 16e22 38716 . . . 4 (   𝑥(𝑥 = 𝑦𝜑)   ,   (𝑥 = 𝑦𝜑)   ▶   [𝑦 / 𝑥]𝜑   )
189, 17exinst 38669 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
198, 18pm3.2i 471 . 2 (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)) ∧ (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
20 impbi 198 . . 3 (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)) → ((∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))))
2120imp 445 . 2 ((([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑)) ∧ (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
2219, 21e0a 38819 1 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  [wsb 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708  df-sb 1879  df-vd1 38606  df-vd2 38614
This theorem is referenced by: (None)
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