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Theorem cbvreuv 2504
Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
cbvralv.1 (x = y → (φψ))
Assertion
Ref Expression
cbvreuv (∃!x A φ∃!y A ψ)
Distinct variable groups:   x,A   y,A   φ,y   ψ,x
Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem cbvreuv
StepHypRef Expression
1 nfv 1394 . 2 yφ
2 nfv 1394 . 2 xψ
3 cbvralv.1 . 2 (x = y → (φψ))
41, 2, 3cbvreu 2500 1 (∃!x A φ∃!y A ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  ∃!wreu 2277
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-cleq 2006  df-clel 2009  df-reu 2282
This theorem is referenced by:  reu8  2705
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