MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbc2ie Structured version   Visualization version   GIF version

Theorem sbc2ie 3834
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
sbc2ie.1 𝐴 ∈ V
sbc2ie.2 𝐵 ∈ V
sbc2ie.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
sbc2ie ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbc2ie
StepHypRef Expression
1 sbc2ie.1 . 2 𝐴 ∈ V
2 sbc2ie.2 . 2 𝐵 ∈ V
3 nfv 1916 . . 3 𝑥𝜓
4 nfv 1916 . . 3 𝑦𝜓
52nfth 1803 . . 3 𝑥 𝐵 ∈ V
6 sbc2ie.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
73, 4, 5, 6sbc2iegf 3833 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
81, 2, 7mp2an 691 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  [wsbc 3758
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-sbc 3759
This theorem is referenced by:  sbc3ie  3836  brfi1uzind  13857  opfi1uzind  13860  wrd2ind  14081  isprs  17536  isdrs  17540  istos  17641  issrg  19253  isslmd  30855  rexrabdioph  39588  rmydioph  39808  rmxdioph  39810  expdiophlem2  39816  2reu8i  43532
  Copyright terms: Public domain W3C validator