Theorem sbmo 2612

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 2280	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤))
2		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = 𝑤
3	2	sblim 2305	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
4	3	sbalv 2168	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
5	4	exbii 1845	. . 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 2538	. . 3 ⊢ (∃*𝑧𝜑 ↔ ∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤))
8	7	sbbii 2074	. 2 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤))
9		df-mo 2538	. 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 1776  [wsb 2062  ∃*wmo 2536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538

This theorem is referenced by: (None)