MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbco3g Structured version   Visualization version   GIF version

Theorem csbco3g 4389
Description: Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
sbcco3g.1 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
csbco3g (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem csbco3g
StepHypRef Expression
1 csbnestg 4387 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐴 / 𝑥𝐵 / 𝑦𝐷)
2 elex 3462 . . . 4 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2905 . . . . 5 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3g.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 3890 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
62, 5syl 17 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
76csbeq1d 3860 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
81, 7eqtrd 2773 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3444  csb 3856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2371  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3446  df-sbc 3741  df-csb 3857
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator