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Theorem sbc2iedv 2986
 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 378 . . 3 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓𝜒))
74, 6sbcied 2950 . 2 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓𝜒))
82, 7sbcied 2950 1 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  Vcvv 2690  [wsbc 2914 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2692  df-sbc 2915 This theorem is referenced by:  dfoprab3  6098
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