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Theorem pm14.12 38101
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 2498 . 2 (∃!𝑥𝜑 → ∃*𝑥𝜑)
2 nfv 1840 . . . 4 𝑦𝜑
32mo3 2506 . . 3 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4 sbsbc 3421 . . . . . 6 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
54anbi2i 729 . . . . 5 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑[𝑦 / 𝑥]𝜑))
65imbi1i 339 . . . 4 (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
762albii 1745 . . 3 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
83, 7bitri 264 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
91, 8sylib 208 1 (∃!𝑥𝜑 → ∀𝑥𝑦((𝜑[𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  [wsb 1877  ∃!weu 2469  ∃*wmo 2470  [wsbc 3417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-sbc 3418
This theorem is referenced by:  pm14.24  38112
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