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Theorem sbc3ie 3858
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
sbc3ie.1 𝐴 ∈ V
sbc3ie.2 𝐵 ∈ V
sbc3ie.3 𝐶 ∈ V
sbc3ie.4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
Assertion
Ref Expression
sbc3ie ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑𝜓)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem sbc3ie
StepHypRef Expression
1 sbc3ie.1 . 2 𝐴 ∈ V
2 sbc3ie.2 . 2 𝐵 ∈ V
3 sbc3ie.3 . . . 4 𝐶 ∈ V
43a1i 11 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝐶 ∈ V)
5 sbc3ie.4 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
653expa 1115 . . 3 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑𝜓))
74, 6sbcied 3817 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ([𝐶 / 𝑧]𝜑𝜓))
81, 2, 7sbc2ie 3855 1 ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3468  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-sbc 3773
This theorem is referenced by:  isdlat  18484  islmod  20707  isslmd  32850  hdmap1fval  41179  hdmapfval  41210  hgmapfval  41269  rmydioph  42313
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