Theorem sbmo 2617

Description: Substitution into an at-most-one quantifier. (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		sbex 2285	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤))
2		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = 𝑤
3	2	sblim 2310	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
4	3	sbalv 2171	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
5	4	exbii 1846	. . 3 ⊢ (∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
6	1, 5	bitri 275	. 2 ⊢ ([𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
7		df-mo 2543	. . 3 ⊢ (∃*𝑧𝜑 ↔ ∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤))
8	7	sbbii 2076	. 2 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤))
9		df-mo 2543	. 2 ⊢ (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
10	6, 8, 9	3bitr4i 303	1 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1777  [wsb 2064  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543

This theorem is referenced by: (None)