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Theorem sb8eu 2039
Description: Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
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 1528 . . . . 5 𝑤(𝜑𝑥 = 𝑧)
21sb8 1856 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 sbbi 1959 . . . . . 6 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4 sb8eu.1 . . . . . . . 8 𝑦𝜑
54nfsb 1946 . . . . . . 7 𝑦[𝑤 / 𝑥]𝜑
6 equsb3 1951 . . . . . . . 8 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
7 nfv 1528 . . . . . . . 8 𝑦 𝑤 = 𝑧
86, 7nfxfr 1474 . . . . . . 7 𝑦[𝑤 / 𝑥]𝑥 = 𝑧
95, 8nfbi 1589 . . . . . 6 𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
103, 9nfxfr 1474 . . . . 5 𝑦[𝑤 / 𝑥](𝜑𝑥 = 𝑧)
11 nfv 1528 . . . . 5 𝑤[𝑦 / 𝑥](𝜑𝑥 = 𝑧)
12 sbequ 1840 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑧)))
1310, 11, 12cbval 1754 . . . 4 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧))
14 equsb3 1951 . . . . . 6 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
1514sblbis 1960 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1615albii 1470 . . . 4 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
172, 13, 163bitri 206 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1817exbii 1605 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
19 df-eu 2029 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
20 df-eu 2029 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
2118, 19, 203bitr4i 212 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 105  wal 1351  wnf 1460  wex 1492  [wsb 1762  ∃!weu 2026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029
This theorem is referenced by:  sb8mo  2040  nfeud  2042  nfeu  2045  cbveu  2050  cbvreu  2701  acexmid  5873
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