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Theorem cbvmo 2505
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 2271 . . 3 (∃𝑥𝜑 ↔ ∃𝑦𝜓)
51, 2, 3cbveu 2504 . . 3 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
64, 5imbi12i 340 . 2 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
7 df-mo 2474 . 2 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
8 df-mo 2474 . 2 (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓))
96, 7, 83bitr4i 292 1 (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1701  wnf 1705  ∃!weu 2469  ∃*wmo 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474
This theorem is referenced by:  dffun6f  5871  opabiotafun  6226  2ndcdisj  21199  cbvdisjf  29271  noprefixmo  31626  phpreu  33064
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