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Theorem sbc3ie 2858
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 9 . . 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 2821 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ([𝐶 / 𝑧]𝜑𝜓))
81, 2, 7sbc2ie 2856 1 ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  Vcvv 2574  [wsbc 2786
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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sbc 2787
This theorem is referenced by: (None)
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