ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  csbco3g Structured version   GIF version

Theorem csbco3g 2898
Description: Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
Hypothesis
Ref Expression
sbcco3g.1 (x = AB = 𝐶)
Assertion
Ref Expression
csbco3g (A 𝑉A / 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 2894 . 2 (A 𝑉A / xB / y𝐷 = A / xB / y𝐷)
2 elex 2560 . . . 4 (A 𝑉A V)
3 nfcvd 2176 . . . . 5 (A V → x𝐶)
4 sbcco3g.1 . . . . 5 (x = AB = 𝐶)
53, 4csbiegf 2884 . . . 4 (A V → A / xB = 𝐶)
62, 5syl 14 . . 3 (A 𝑉A / xB = 𝐶)
76csbeq1d 2852 . 2 (A 𝑉A / xB / y𝐷 = 𝐶 / y𝐷)
81, 7eqtrd 2069 1 (A 𝑉A / xB / y𝐷 = 𝐶 / y𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  Vcvv 2551  csb 2846
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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator