MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvrexv2 Structured version   Visualization version   GIF version

Theorem cbvrexv2 3943
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvralv2.1 (𝑥 = 𝑦 → (𝜓𝜒))
cbvralv2.2 (𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
cbvrexv2 (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 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 2903 . 2 𝑦𝐴
2 nfcv 2903 . 2 𝑥𝐵
3 nfv 1917 . 2 𝑦𝜓
4 nfv 1917 . 2 𝑥𝜒
5 cbvralv2.2 . 2 (𝑥 = 𝑦𝐴 = 𝐵)
6 cbvralv2.1 . 2 (𝑥 = 𝑦 → (𝜓𝜒))
71, 2, 3, 4, 5, 6cbvrexcsf 3939 1 (∃𝑥𝐴 𝜓 ↔ ∃𝑦𝐵 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-sbc 3778  df-csb 3894
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator