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Theorem cbveu 2224
 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 2222 . 2 (∃!xφ∃!y[y / x]φ)
3 cbveu.2 . . . 4 xψ
4 cbveu.3 . . . 4 (x = y → (φψ))
53, 4sbie 2038 . . 3 ([y / x]φψ)
65eubii 2213 . 2 (∃!y[y / x]φ∃!yψ)
72, 6bitri 240 1 (∃!xφ∃!yψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  [wsb 1648  ∃!weu 2204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208 This theorem is referenced by:  cbvmo  2241  cbvreu  2833  cbvreucsf  3200  tz6.12-1  5344
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