Theorem sbmo 2642

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 2316	. . 3 ⊢ ([𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤))
2		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = 𝑤
3	2	sblim 2340	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
4	3	sbalv 2205	. . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
5	4	exbii 1878	. . 3 ⊢ (∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
6	1, 5	bitri 278	. 2 ⊢ ([𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
7		dfmo 2568	. . 3 ⊢ (∃*𝑧𝜑 ↔ ∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤))
8	7	sbbii 2110	. 2 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ [𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤))
9		dfmo 2568	. 2 ⊢ (∃*𝑧[𝑦 / 𝑥]𝜑 ↔ ∃𝑤∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤))
10	6, 8, 9	3bitr4i 306	1 ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1809  [wsb 2096  ∃*wmo 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567

This theorem is referenced by: (None)