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Theorem cbvmo 1983
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 1681 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
51, 2, 3cbveu 1967 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
64, 5imbi12i 237 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
7 df-mo 1947 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8 df-mo 1947 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
96, 7, 83bitr4i 210 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wnf 1390  wex 1422  ∃!weu 1943  ∃*wmo 1944
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947
This theorem is referenced by:  dffun6f  4966
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