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Theorem sbciegf 3795
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 1797 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 sbciegft 3794 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
51, 3, 4mp3an23 1450 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wnf 1785  wcel 2115  [wsbc 3758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-sbc 3759
This theorem is referenced by:  sbcieg  3796  rexsngf  4595  ralsngf  4596  opelopabgf  5414  opelopabf  5419  eqerlem  8319  bnj919  32095  bnj1464  32173  bnj1123  32315  bnj1373  32359  poimirlem25  35027  sbccomieg  39650  aomclem6  39919  fveqsb  41077
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