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Theorem sbc2iedv 3473
 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 11 . 2 (𝜑𝐴 ∈ V)
3 sbc2iedv.2 . . . 4 𝐵 ∈ V
43a1i 11 . . 3 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
5 sbc2iedv.3 . . . 4 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
65impl 648 . . 3 (((𝜑𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓𝜒))
74, 6sbcied 3439 . 2 ((𝜑𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓𝜒))
82, 7sbcied 3439 1 (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  dfoprab3  7115  sbcie2s  15744  ismnddef  17119  sdclem1  32709
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