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Theorem cbvmo 2017
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvmo.1 𝑦𝜑
cbvmo.2 𝑥𝜓
cbvmo.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvmo (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)

Proof of Theorem cbvmo
StepHypRef Expression
1 cbvmo.1 . . . 4 𝑦𝜑
2 cbvmo.2 . . . 4 𝑥𝜓
3 cbvmo.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvex 1714 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
51, 2, 3cbveu 2001 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
64, 5imbi12i 238 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
7 df-mo 1981 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8 df-mo 1981 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
96, 7, 83bitr4i 211 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1421  wex 1453  ∃!weu 1977  ∃*wmo 1978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981
This theorem is referenced by:  dffun6f  5106
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