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Theorem sbc2iegf 2893
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 107 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpl 107 . . . 4 ((𝐵𝑊𝑥 = 𝐴) → 𝐵𝑊)
3 sbc2iegf.4 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
43adantll 460 . . . 4 (((𝐵𝑊𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜑𝜓))
5 nfv 1462 . . . 4 𝑦(𝐵𝑊𝑥 = 𝐴)
6 sbc2iegf.2 . . . . 5 𝑦𝜓
76a1i 9 . . . 4 ((𝐵𝑊𝑥 = 𝐴) → Ⅎ𝑦𝜓)
82, 4, 5, 7sbciedf 2858 . . 3 ((𝐵𝑊𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑𝜓))
98adantll 460 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑𝜓))
10 nfv 1462 . . 3 𝑥 𝐴𝑉
11 sbc2iegf.3 . . 3 𝑥 𝐵𝑊
1210, 11nfan 1498 . 2 𝑥(𝐴𝑉𝐵𝑊)
13 sbc2iegf.1 . . 3 𝑥𝜓
1413a1i 9 . 2 ((𝐴𝑉𝐵𝑊) → Ⅎ𝑥𝜓)
151, 9, 12, 14sbciedf 2858 1 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wnf 1390  wcel 1434  [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:  sbc2ie  2894  opelopabaf  4056
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