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Theorem sb8euh 1939
Description: Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)
Hypothesis
Ref Expression
sb8euh.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
sb8euh (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8euh
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-17 1435 . . . . 5 ((𝜑𝑥 = 𝑧) → ∀𝑤(𝜑𝑥 = 𝑧))
21sb8h 1750 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 sbbi 1849 . . . . . 6 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4 sb8euh.1 . . . . . . . 8 (𝜑 → ∀𝑦𝜑)
54hbsb 1839 . . . . . . 7 ([𝑤 / 𝑥]𝜑 → ∀𝑦[𝑤 / 𝑥]𝜑)
6 equsb3 1841 . . . . . . . 8 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
7 ax-17 1435 . . . . . . . 8 (𝑤 = 𝑧 → ∀𝑦 𝑤 = 𝑧)
86, 7hbxfrbi 1377 . . . . . . 7 ([𝑤 / 𝑥]𝑥 = 𝑧 → ∀𝑦[𝑤 / 𝑥]𝑥 = 𝑧)
95, 8hbbi 1456 . . . . . 6 (([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧) → ∀𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
103, 9hbxfrbi 1377 . . . . 5 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) → ∀𝑦[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
11 ax-17 1435 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥 = 𝑧) → ∀𝑤[𝑦 / 𝑥](𝜑𝑥 = 𝑧))
12 sbequ 1737 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑧)))
1310, 11, 12cbvalh 1652 . . . 4 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧))
14 equsb3 1841 . . . . . 6 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
1514sblbis 1850 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1615albii 1375 . . . 4 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
172, 13, 163bitri 199 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1817exbii 1512 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
19 df-eu 1919 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
20 df-eu 1919 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
2118, 19, 203bitr4i 205 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257  wex 1397  [wsb 1661  ∃!weu 1916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919
This theorem is referenced by:  eu1  1941
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