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Theorem pm14.12 43889
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 2567 . 2 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 nfv 1909 . . . 4 𝑦𝜑
32mo3 2553 . . 3 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4 sbsbc 3782 . . . . . 6 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
54anbi2i 621 . . . . 5 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑[𝑦 / 𝑥]𝜑))
65imbi1i 348 . . . 4 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
762albii 1814 . . 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 394  wal 1531  [wsb 2059  ∃*wmo 2527  ∃!weu 2557  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-sbc 3779
This theorem is referenced by:  pm14.24  43900
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