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Theorem sbc2iegf 2853
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 106 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpl 106 . . . 4 ((𝐵𝑊𝑥 = 𝐴) → 𝐵𝑊)
3 sbc2iegf.4 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
43adantll 453 . . . 4 (((𝐵𝑊𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜑𝜓))
5 nfv 1435 . . . 4 𝑦(𝐵𝑊𝑥 = 𝐴)
6 sbc2iegf.2 . . . . 5 𝑦𝜓
76a1i 9 . . . 4 ((𝐵𝑊𝑥 = 𝐴) → Ⅎ𝑦𝜓)
82, 4, 5, 7sbciedf 2818 . . 3 ((𝐵𝑊𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑𝜓))
98adantll 453 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑𝜓))
10 nfv 1435 . . 3 𝑥 𝐴𝑉
11 sbc2iegf.3 . . 3 𝑥 𝐵𝑊
1210, 11nfan 1471 . 2 𝑥(𝐴𝑉𝐵𝑊)
13 sbc2iegf.1 . . 3 𝑥𝜓
1413a1i 9 . 2 ((𝐴𝑉𝐵𝑊) → Ⅎ𝑥𝜓)
151, 9, 12, 14sbciedf 2818 1 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wnf 1363  wcel 1407  [wsbc 2784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-sbc 2785
This theorem is referenced by:  sbc2ie  2854  opelopabaf  4035
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