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Theorem cbvmo 1940
 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 1639 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
51, 2, 3cbveu 1924 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
64, 5imbi12i 228 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
7 df-mo 1904 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8 df-mo 1904 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
96, 7, 83bitr4i 201 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1349  ∃wex 1381  ∃!weu 1900  ∃*wmo 1901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904 This theorem is referenced by:  dffun6f  4915
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