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Theorem sbc2iedv 2953
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
Hypotheses
Ref Expression
sbc2iedv.1 𝐴 ∈ V
sbc2iedv.2 𝐵 ∈ V
sbc2iedv.3 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
Assertion
Ref Expression
sbc2iedv (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜑,𝑥,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbc2iedv
StepHypRef Expression
1 sbc2iedv.1 . . 3 𝐴 ∈ V
21a1i 9 . 2 (𝜑𝐴 ∈ V)
3 sbc2iedv.2 . . . 4 𝐵 ∈ V
43a1i 9 . . 3 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
5 sbc2iedv.3 . . . 4 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
65impl 377 . . 3 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓𝜒))
74, 6sbcied 2917 . 2 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓𝜒))
82, 7sbcied 2917 1 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  Vcvv 2660  [wsbc 2882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by:  dfoprab3  6057
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