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Theorem sb8eu 2624
Description: Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)
Hypothesis
Ref Expression
sb8eu.1 𝑦𝜑
Assertion
Ref Expression
sb8eu (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8eu
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 2009 . . . . 5 𝑤(𝜑𝑥 = 𝑧)
21sb8 2515 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 equsb3 2524 . . . . . 6 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
43sblbis 2495 . . . . 5 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑𝑤 = 𝑧))
54albii 1914 . . . 4 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑤([𝑤 / 𝑥]𝜑𝑤 = 𝑧))
6 sb8eu.1 . . . . . . 7 𝑦𝜑
76nfsb 2534 . . . . . 6 𝑦[𝑤 / 𝑥]𝜑
8 nfv 2009 . . . . . 6 𝑦 𝑤 = 𝑧
97, 8nfbi 2002 . . . . 5 𝑦([𝑤 / 𝑥]𝜑𝑤 = 𝑧)
10 nfv 2009 . . . . 5 𝑤([𝑦 / 𝑥]𝜑𝑦 = 𝑧)
11 sbequ 2467 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
12 equequ1 2122 . . . . . 6 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
1311, 12bibi12d 336 . . . . 5 (𝑤 = 𝑦 → (([𝑤 / 𝑥]𝜑𝑤 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
149, 10, 13cbval 2376 . . . 4 (∀𝑤([𝑤 / 𝑥]𝜑𝑤 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
152, 5, 143bitri 288 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1615exbii 1943 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
17 eu6 2589 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
18 eu6 2589 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1916, 17, 183bitr4i 294 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1650  wex 1874  wnf 1878  [wsb 2062  ∃!weu 2581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582
This theorem is referenced by:  sb8mo  2625  cbveu  2626  eu1  2631  cbvreu  3317
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