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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pm14.12 | Structured version Visualization version GIF version | ||
| Description: Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.) | 
| Ref | Expression | 
|---|---|
| pm14.12 | ⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eumo 2577 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
| 2 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
| 3 | 2 | mo3 2563 | . . 3 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | 
| 4 | sbsbc 3791 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
| 5 | 4 | anbi2i 623 | . . . . 5 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) | 
| 6 | 5 | imbi1i 349 | . . . 4 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | 
| 7 | 6 | 2albii 1819 | . . 3 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | 
| 8 | 3, 7 | bitri 275 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | 
| 9 | 1, 8 | sylib 218 | 1 ⊢ (∃!𝑥𝜑 → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1537 [wsb 2063 ∃*wmo 2537 ∃!weu 2567 [wsbc 3787 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: pm14.24 44456 | 
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