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Theorem cbvrexv2 2886
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
cbvralv2.1 (x = y → (ψχ))
cbvralv2.2 (x = yA = B)
Assertion
Ref Expression
cbvrexv2 (x A ψy B χ)
Distinct variable groups:   y,A   ψ,y   x,B   χ,x
Allowed substitution hints:   ψ(x)   χ(y)   A(x)   B(y)

Proof of Theorem cbvrexv2
StepHypRef Expression
1 nfcv 2156 . 2 yA
2 nfcv 2156 . 2 xB
3 nfv 1398 . 2 yψ
4 nfv 1398 . 2 xχ
5 cbvralv2.2 . 2 (x = yA = B)
6 cbvralv2.1 . 2 (x = y → (ψχ))
71, 2, 3, 4, 5, 6cbvrexcsf 2882 1 (x A ψy B χ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1226  wrex 2281
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rex 2286  df-sbc 2738  df-csb 2826
This theorem is referenced by: (None)
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