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Theorem sbciegf 3831
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
sbciegf.1 𝑥𝜓
sbciegf.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
sbciegf (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem sbciegf
StepHypRef Expression
1 sbciegf.1 . 2 𝑥𝜓
2 sbciegf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32ax-gen 1792 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 sbciegft 3829 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
51, 3, 4mp3an23 1452 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wnf 1780  wcel 2106  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792
This theorem is referenced by:  sbciegOLD  3833  rexsngf  4677  ralsngf  4678  opelopabgf  5550  opelopabf  5555  eqerlem  8779  bnj919  34760  bnj1464  34837  bnj1123  34979  bnj1373  35023  poimirlem25  37632  sbccomieg  42781  aomclem6  43048  fveqsb  44449
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