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Theorem sbc2ie 2952
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
sbc2ie.1 𝐴 ∈ V
sbc2ie.2 𝐵 ∈ V
sbc2ie.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
sbc2ie ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbc2ie
StepHypRef Expression
1 sbc2ie.1 . 2 𝐴 ∈ V
2 sbc2ie.2 . 2 𝐵 ∈ V
3 nfv 1493 . . 3 𝑥𝜓
4 nfv 1493 . . 3 𝑦𝜓
52nfth 1425 . . 3 𝑥 𝐵 ∈ V
6 sbc2ie.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
73, 4, 5, 6sbc2iegf 2951 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
81, 2, 7mp2an 422 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wcel 1465  Vcvv 2660  [wsbc 2882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by:  sbc3ie  2954
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