MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbciegf Structured version   Visualization version   GIF version

Theorem sbciegf 3617
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 1869 . 2 𝑥(𝑥 = 𝐴 → (𝜑𝜓))
4 sbciegft 3616 . 2 ((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))
51, 3, 4mp3an23 1563 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628   = wceq 1630  wnf 1855  wcel 2144  [wsbc 3585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-v 3351  df-sbc 3586
This theorem is referenced by:  sbcieg  3618  opelopabgf  5128  opelopabf  5133  eqerlem  7929  iunxsngf  29707  bnj919  31169  bnj1464  31246  bnj1123  31386  bnj1373  31430  poimirlem25  33760  sbccomieg  37876  aomclem6  38148  fveqsb  39176  rexsngf  39735
  Copyright terms: Public domain W3C validator