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Theorem sbc2iegf 2951
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
sbc2iegf.1 𝑥𝜓
sbc2iegf.2 𝑦𝜓
sbc2iegf.3 𝑥 𝐵𝑊
sbc2iegf.4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
sbc2iegf ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝑉   𝑦,𝑊
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐵(𝑥)   𝑉(𝑦)   𝑊(𝑥)

Proof of Theorem sbc2iegf
StepHypRef Expression
1 simpl 108 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpl 108 . . . 4 ((𝐵𝑊𝑥 = 𝐴) → 𝐵𝑊)
3 sbc2iegf.4 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
43adantll 467 . . . 4 (((𝐵𝑊𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜑𝜓))
5 nfv 1493 . . . 4 𝑦(𝐵𝑊𝑥 = 𝐴)
6 sbc2iegf.2 . . . . 5 𝑦𝜓
76a1i 9 . . . 4 ((𝐵𝑊𝑥 = 𝐴) → Ⅎ𝑦𝜓)
82, 4, 5, 7sbciedf 2916 . . 3 ((𝐵𝑊𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑𝜓))
98adantll 467 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑𝜓))
10 nfv 1493 . . 3 𝑥 𝐴𝑉
11 sbc2iegf.3 . . 3 𝑥 𝐵𝑊
1210, 11nfan 1529 . 2 𝑥(𝐴𝑉𝐵𝑊)
13 sbc2iegf.1 . . 3 𝑥𝜓
1413a1i 9 . 2 ((𝐴𝑉𝐵𝑊) → Ⅎ𝑥𝜓)
151, 9, 12, 14sbciedf 2916 1 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wnf 1421  wcel 1465  [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:  sbc2ie  2952  opelopabaf  4165
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