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Theorem cbveu 1999
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
cbveu.1 𝑦𝜑
cbveu.2 𝑥𝜓
cbveu.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbveu (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Proof of Theorem cbveu
StepHypRef Expression
1 cbveu.1 . . 3 𝑦𝜑
21sb8eu 1988 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3 cbveu.2 . . . 4 𝑥𝜓
4 cbveu.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4sbie 1747 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
65eubii 1984 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
72, 6bitri 183 1 (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1419  [wsb 1718  ∃!weu 1975
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978
This theorem is referenced by:  cbvmo  2015  cbvreu  2627  cbvreucsf  3032  tz6.12f  5416  f1ompt  5537  climeu  11016
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