Theorem sbmo 2112

Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
sbmo	⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbmo

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1550	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = 𝑤
2	1	sblim 1984	. . . . 5 ⊢ ([𝑦 / 𝑥]((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
3		sban 1982	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑))
4	3	imbi1i 238	. . . . 5 ⊢ (([𝑦 / 𝑥](𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
5		sbcom2 2014	. . . . . . 7 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑)
6	5	anbi2i 457	. . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑))
7	6	imbi1i 238	. . . . 5 ⊢ ((([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
8	2, 4, 7	3bitri 206	. . . 4 ⊢ ([𝑦 / 𝑥]((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ (([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
9	8	sbalv 2032	. . 3 ⊢ ([𝑦 / 𝑥]∀𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ∀𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
10	9	sbalv 2032	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧∀𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤) ↔ ∀𝑧∀𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
11		nfv 1550	. . . 4 ⊢ Ⅎ𝑤𝜑
12	11	mo3 2107	. . 3 ⊢ (∃*𝑧𝜑 ↔ ∀𝑧∀𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
13	12	sbbii 1787	. 2 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∀𝑧∀𝑤((𝜑 ∧ [𝑤 / 𝑧]𝜑) → 𝑧 = 𝑤))
14		nfv 1550	. . 3 ⊢ Ⅎ𝑤[𝑦 / 𝑥]𝜑
15	14	mo3 2107	. 2 ⊢ (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧∀𝑤(([𝑦 / 𝑥]𝜑 ∧ [𝑤 / 𝑧][𝑦 / 𝑥]𝜑) → 𝑧 = 𝑤))
16	10, 13, 15	3bitr4i 212	1 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1370   = wceq 1372  [wsb 1784  ∃*wmo 2054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057

This theorem is referenced by: (None)