New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  cbvrexv2 GIF version

Theorem cbvrexv2 3203
 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 2489 . 2 yA
2 nfcv 2489 . 2 xB
3 nfv 1619 . 2 yψ
4 nfv 1619 . 2 xχ
5 cbvralv2.2 . 2 (x = yA = B)
6 cbvralv2.1 . 2 (x = y → (ψχ))
71, 2, 3, 4, 5, 6cbvrexcsf 3199 1 (x A ψy B χ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∃wrex 2615 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  ax-ext 2334 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-sbc 3047  df-csb 3137 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator