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Theorem sb8eulem 2662
Description: Lemma. Factor out the common proof skeleton of sb8euv 2663 and sb8eu 2664. 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.) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023.)
Hypothesis
Ref Expression
sb8eulem.nfsb 𝑦[𝑤 / 𝑥]𝜑
Assertion
Ref Expression
sb8eulem (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑦,𝑤   𝜑,𝑤   𝑥,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8eulem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . . 5 𝑤(𝜑𝑥 = 𝑧)
21sb8v 2365 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 equsb3 2107 . . . . . 6 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
43sblbis 2316 . . . . 5 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑𝑤 = 𝑧))
54albii 1821 . . . 4 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑤([𝑤 / 𝑥]𝜑𝑤 = 𝑧))
6 sb8eulem.nfsb . . . . . 6 𝑦[𝑤 / 𝑥]𝜑
7 nfv 1915 . . . . . 6 𝑦 𝑤 = 𝑧
86, 7nfbi 1904 . . . . 5 𝑦([𝑤 / 𝑥]𝜑𝑤 = 𝑧)
9 nfv 1915 . . . . 5 𝑤([𝑦 / 𝑥]𝜑𝑦 = 𝑧)
10 sbequ 2089 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
11 equequ1 2032 . . . . . 6 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
1210, 11bibi12d 349 . . . . 5 (𝑤 = 𝑦 → (([𝑤 / 𝑥]𝜑𝑤 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
138, 9, 12cbvalv1 2353 . . . 4 (∀𝑤([𝑤 / 𝑥]𝜑𝑤 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
142, 5, 133bitri 300 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1514exbii 1849 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
16 eu6 2637 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
17 eu6 2637 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1815, 16, 173bitr4i 306 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  wex 1781  wnf 1785  [wsb 2069  ∃!weu 2631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632
This theorem is referenced by:  sb8euv  2663  sb8eu  2664
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