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Theorem sbc2ie 3861
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Gino Giotto, 12-Oct-2024.)
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 . . . 4 𝐵 ∈ V
32a1i 11 . . 3 (𝑥 = 𝐴𝐵 ∈ V)
4 sbc2ie.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
53, 4sbcied 3823 . 2 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑𝜓))
61, 5sbcie 3821 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  [wsbc 3778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sbc 3779
This theorem is referenced by:  sbc3ie  3864  brfi1uzind  14459  opfi1uzind  14462  wrd2ind  14673  isprs  18250  isdrs  18254  istos  18371  issrg  20011  isslmd  32347  rexrabdioph  41532  rmydioph  41753  rmxdioph  41755  expdiophlem2  41761  2reu8i  45821
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