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Theorem sbc2iegf 3654
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 468 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
2 simpl 468 . . . 4 ((𝐵𝑊𝑥 = 𝐴) → 𝐵𝑊)
3 sbc2iegf.4 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
43adantll 693 . . . 4 (((𝐵𝑊𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜑𝜓))
5 nfv 1995 . . . 4 𝑦(𝐵𝑊𝑥 = 𝐴)
6 sbc2iegf.2 . . . . 5 𝑦𝜓
76a1i 11 . . . 4 ((𝐵𝑊𝑥 = 𝐴) → Ⅎ𝑦𝜓)
82, 4, 5, 7sbciedf 3623 . . 3 ((𝐵𝑊𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑𝜓))
98adantll 693 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜑𝜓))
10 nfv 1995 . . 3 𝑥 𝐴𝑉
11 sbc2iegf.3 . . 3 𝑥 𝐵𝑊
1210, 11nfan 1980 . 2 𝑥(𝐴𝑉𝐵𝑊)
13 sbc2iegf.1 . . 3 𝑥𝜓
1413a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → Ⅎ𝑥𝜓)
151, 9, 12, 14sbciedf 3623 1 ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wnf 1856  wcel 2145  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by:  sbc2ie  3655  opelopabaf  5132  elmptrab  21851  vtocl2d  29654
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