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Theorem sbc3ie 3648
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 1112 . . 3 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑𝜓))
74, 6sbcied 3613 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → ([𝐶 / 𝑧]𝜑𝜓))
81, 2, 7sbc2ie 3646 1 ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  [wsbc 3576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342  df-sbc 3577
This theorem is referenced by:  isdlat  17394  islmod  19069  isslmd  30064  hdmap1fval  37588  hdmapfval  37621  hgmapfval  37680  rmydioph  38083
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