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Theorem sbc2ie 3730
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 2015 . . 3 𝑥𝜓
4 nfv 2015 . . 3 𝑦𝜓
52nfth 1902 . . 3 𝑥 𝐵 ∈ V
6 sbc2ie.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
73, 4, 5, 6sbc2iegf 3729 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
81, 2, 7mp2an 685 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  Vcvv 3414  [wsbc 3662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-12 2222  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416  df-sbc 3663
This theorem is referenced by:  sbc3ie  3732  brfi1uzind  13569  opfi1uzind  13572  wrd2ind  13814  wrd2indOLD  13815  isprs  17283  isdrs  17287  istos  17388  issrg  18861  isslmd  30300  rexrabdioph  38202  rmydioph  38424  rmxdioph  38426  expdiophlem2  38432
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