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Theorem sbc2iedv 2895
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 372 . . 3 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓𝜒))
74, 6sbcied 2859 . 2 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓𝜒))
82, 7sbcied 2859 1 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wcel 1434  Vcvv 2610  [wsbc 2824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2825
This theorem is referenced by:  dfoprab3  5868
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