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Theorem pm14.12 42039
Description: Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
pm14.12 (∃!𝑥𝜑 → ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Distinct variable groups:   𝜑,𝑦   𝑥,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem pm14.12
StepHypRef Expression
1 eumo 2578 . 2 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 nfv 1917 . . . 4 𝑦𝜑
32mo3 2564 . . 3 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4 sbsbc 3720 . . . . . 6 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
54anbi2i 623 . . . . 5 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑[𝑦 / 𝑥]𝜑))
65imbi1i 350 . . . 4 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
762albii 1823 . . 3 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
83, 7bitri 274 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
91, 8sylib 217 1 (∃!𝑥𝜑 → ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1537  [wsb 2067  ∃*wmo 2538  ∃!weu 2568  [wsbc 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-sbc 3717
This theorem is referenced by:  pm14.24  42050
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