MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcied2 Structured version   Visualization version   GIF version

Theorem sbcied2 3456
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied2.1 (𝜑𝐴𝑉)
sbcied2.2 (𝜑𝐴 = 𝐵)
sbcied2.3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
Assertion
Ref Expression
sbcied2 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem sbcied2
StepHypRef Expression
1 sbcied2.1 . 2 (𝜑𝐴𝑉)
2 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
3 sbcied2.2 . . . 4 (𝜑𝐴 = 𝐵)
42, 3sylan9eqr 2677 . . 3 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐵)
5 sbcied2.3 . . 3 ((𝜑𝑥 = 𝐵) → (𝜓𝜒))
64, 5syldan 487 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
71, 6sbcied 3455 1 (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  [wsbc 3418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188  df-sbc 3419
This theorem is referenced by:  iscat  16257  sectffval  16334  issubc  16419  isfunc  16448  ismgm  17167  issgrp  17209  isnsg  17547  isring  18475  islbs  18998  isdomn  19216  isassa  19237  opsrval  19396  isuhgr  25858  isushgr  25859  isupgr  25882  isumgr  25892  isuspgr  25947  isusgr  25948  isfrgr  26995  isrng  41180
  Copyright terms: Public domain W3C validator