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Theorem sbc2ie 3874
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 GG, 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 3837 . 2 (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑𝜓))
61, 5sbcie 3835 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792
This theorem is referenced by:  sbc3ie  3877  brfi1uzind  14544  opfi1uzind  14547  wrd2ind  14758  isprs  18354  isdrs  18359  istos  18476  issrg  20206  isslmd  33191  rexrabdioph  42782  rmydioph  43003  rmxdioph  43005  expdiophlem2  43011  2reu8i  47063
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