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Theorem sb8eu 1961
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 1466 . . . . 5 𝑤(𝜑𝑥 = 𝑧)
21sb8 1784 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 sbbi 1881 . . . . . 6 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4 sb8eu.1 . . . . . . . 8 𝑦𝜑
54nfsb 1870 . . . . . . 7 𝑦[𝑤 / 𝑥]𝜑
6 equsb3 1873 . . . . . . . 8 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
7 nfv 1466 . . . . . . . 8 𝑦 𝑤 = 𝑧
86, 7nfxfr 1408 . . . . . . 7 𝑦[𝑤 / 𝑥]𝑥 = 𝑧
95, 8nfbi 1526 . . . . . 6 𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
103, 9nfxfr 1408 . . . . 5 𝑦[𝑤 / 𝑥](𝜑𝑥 = 𝑧)
11 nfv 1466 . . . . 5 𝑤[𝑦 / 𝑥](𝜑𝑥 = 𝑧)
12 sbequ 1768 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑧)))
1310, 11, 12cbval 1684 . . . 4 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧))
14 equsb3 1873 . . . . . 6 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
1514sblbis 1882 . . . . 5 ([𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1615albii 1404 . . . 4 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
172, 13, 163bitri 204 . . 3 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1817exbii 1541 . 2 (∃𝑧𝑥(𝜑𝑥 = 𝑧) ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
19 df-eu 1951 . 2 (∃!𝑥𝜑 ↔ ∃𝑧𝑥(𝜑𝑥 = 𝑧))
20 df-eu 1951 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑧𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
2118, 19, 203bitr4i 210 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1287  wnf 1394  wex 1426  [wsb 1692  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951
This theorem is referenced by:  sb8mo  1962  nfeud  1964  nfeu  1967  cbveu  1972  cbvreu  2588  acexmid  5633
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