MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  csbie2g Structured version   Visualization version   GIF version

Theorem csbie2g 3722
Description: Conversion of implicit substitution to explicit class substitution. This version of csbie 3717 avoids a disjointness condition on 𝑥, 𝐴 and 𝑥, 𝐷 by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
Hypotheses
Ref Expression
csbie2g.1 (𝑥 = 𝑦𝐵 = 𝐶)
csbie2g.2 (𝑦 = 𝐴𝐶 = 𝐷)
Assertion
Ref Expression
csbie2g (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑦,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem csbie2g
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3692 . 2 𝐴 / 𝑥𝐵 = {𝑧[𝐴 / 𝑥]𝑧𝐵}
2 csbie2g.1 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
32eleq2d 2830 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
4 csbie2g.2 . . . . 5 (𝑦 = 𝐴𝐶 = 𝐷)
54eleq2d 2830 . . . 4 (𝑦 = 𝐴 → (𝑧𝐶𝑧𝐷))
63, 5sbcie2g 3630 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐵𝑧𝐷))
76abbi1dv 2886 . 2 (𝐴𝑉 → {𝑧[𝐴 / 𝑥]𝑧𝐵} = 𝐷)
81, 7syl5eq 2811 1 (𝐴𝑉𝐴 / 𝑥𝐵 = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  {cab 2751  [wsbc 3596  csb 3691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-v 3352  df-sbc 3597  df-csb 3692
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator