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| Description: Substitution into an at-most-one quantifier. (Contributed by Jeff Madsen, 2-Sep-2009.) | 
| Ref | Expression | 
|---|---|
| sbmo | ⊢ ([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbex 2281 | . . 3 ⊢ ([𝑦 / 𝑥]∃𝑤∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∃𝑤[𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤)) | |
| 2 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑥 𝑧 = 𝑤 | |
| 3 | 2 | sblim 2306 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 → 𝑧 = 𝑤) ↔ ([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤)) | 
| 4 | 3 | sbalv 2170 | . . . 4 ⊢ ([𝑦 / 𝑥]∀𝑧(𝜑 → 𝑧 = 𝑤) ↔ ∀𝑧([𝑦 / 𝑥]𝜑 → 𝑧 = 𝑤)) | 
| 5 | 4 | exbii 1848 | . . 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 2540 | . . 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 2540 | . 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) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 ∃wex 1779 [wsb 2064 ∃*wmo 2538 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 | 
| This theorem is referenced by: (None) | 
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