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Theorem cbvrexv2 2907
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 2175 . 2 yA
2 nfcv 2175 . 2 xB
3 nfv 1418 . 2 yψ
4 nfv 1418 . 2 xχ
5 cbvralv2.2 . 2 (x = yA = B)
6 cbvralv2.1 . 2 (x = y → (ψχ))
71, 2, 3, 4, 5, 6cbvrexcsf 2903 1 (x A ψy B χ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wrex 2301
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-sbc 2759  df-csb 2847
This theorem is referenced by: (None)
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