Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elimhyps Structured version   Visualization version   GIF version

Theorem elimhyps 33155
 Description: A version of elimhyp 3999 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
Hypothesis
Ref Expression
elimhyps.1 [𝐵 / 𝑥]𝜑
Assertion
Ref Expression
elimhyps [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑

Proof of Theorem elimhyps
StepHypRef Expression
1 sbceq1a 3317 . . 3 (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
2 dfsbcq 3308 . . 3 (𝐵 = if(𝜑, 𝑥, 𝐵) → ([𝐵 / 𝑥]𝜑[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
3 elimhyps.1 . . 3 [𝐵 / 𝑥]𝜑
41, 2, 3elimhyp 3999 . 2 [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
5 biid 249 . . 3 (𝜑𝜑)
6 ifbi 3960 . . 3 ((𝜑𝜑) → if(𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵))
7 dfsbcq 3308 . . . 4 (if(𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵) → ([if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
87bicomd 211 . . 3 (if(𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵) → ([if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑))
95, 6, 8mp2b 10 . 2 ([if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑)
104, 9mpbir 219 1 [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 194   = wceq 1474  [wsbc 3306  ifcif 3939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-sbc 3307  df-if 3940 This theorem is referenced by:  renegclALT  33157
 Copyright terms: Public domain W3C validator