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Theorem sb8mo 1962
Description: Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
sb8eu.1 𝑦𝜑
Assertion
Ref Expression
sb8mo (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8mo
StepHypRef Expression
1 sb8eu.1 . . . 4 𝑦𝜑
21sb8e 1785 . . 3 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
31sb8eu 1961 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
42, 3imbi12i 237 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
5 df-mo 1952 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6 df-mo 1952 . 2 (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ (∃𝑦[𝑦 / 𝑥]𝜑 → ∃!𝑦[𝑦 / 𝑥]𝜑))
74, 5, 63bitr4i 210 1 (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wnf 1394  wex 1426  [wsb 1692  ∃!weu 1948  ∃*wmo 1949
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  df-mo 1952
This theorem is referenced by: (None)
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