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Theorem sb8eulem 2686
Description: Lemma. Factor out the common proof skeleton of sb8euv 2687 and sb8eu 2688. 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.)
Hypotheses
Ref Expression
sb8eulem.nfsb 𝑦[𝑤 / 𝑥]𝜑
sb8eulem.sbequ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 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 2013 . . . . 5 𝑤(𝜑𝑥 = 𝑧)
21sb8v 2376 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 equsb3v 2347 . . . . . 6 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
43sblbisv 2342 . . . . 5 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑𝑤 = 𝑧))
54albii 1918 . . . 4 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑤([𝑤 / 𝑥]𝜑𝑤 = 𝑧))
6 sb8eulem.nfsb . . . . . 6 𝑦[𝑤 / 𝑥]𝜑
7 nfv 2013 . . . . . 6 𝑦 𝑤 = 𝑧
86, 7nfbi 2006 . . . . 5 𝑦([𝑤 / 𝑥]𝜑𝑤 = 𝑧)
9 nfv 2013 . . . . 5 𝑤([𝑦 / 𝑥]𝜑𝑦 = 𝑧)
10 sb8eulem.sbequ . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
11 equequ1 2129 . . . . . 6 (𝑤 = 𝑦 → (𝑤 = 𝑧𝑦 = 𝑧))
1210, 11bibi12d 337 . . . . 5 (𝑤 = 𝑦 → (([𝑤 / 𝑥]𝜑𝑤 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧)))
138, 9, 12cbvalv1 2367 . . . 4 (∀𝑤([𝑤 / 𝑥]𝜑𝑤 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
142, 5, 133bitri 289 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1514exbii 1947 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
16 eu6 2645 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
17 eu6 2645 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1815, 16, 173bitr4i 295 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1654  wex 1878  wnf 1882  [wsb 2067  ∃!weu 2639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640
This theorem is referenced by:  sb8euv  2687  sb8eu  2688
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