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

Theorem csbco3g 3026
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 (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 3022 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐴 / 𝑥𝐵 / 𝑦𝐷)
2 elex 2669 . . . 4 (𝐴𝑉𝐴 ∈ V)
3 nfcvd 2257 . . . . 5 (𝐴 ∈ V → 𝑥𝐶)
4 sbcco3g.1 . . . . 5 (𝑥 = 𝐴𝐵 = 𝐶)
53, 4csbiegf 3011 . . . 4 (𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐶)
62, 5syl 14 . . 3 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐶)
76csbeq1d 2979 . 2 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
81, 7eqtrd 2148 1 (𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐷 = 𝐶 / 𝑦𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  Vcvv 2658  csb 2973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator