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

Theorem dedths 34566
Description: A version of weak deduction theorem dedth 4172 using explicit substitution. (Contributed by NM, 15-Jun-2019.)
Hypothesis
Ref Expression
dedths.1 [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓
Assertion
Ref Expression
dedths (𝜑𝜓)

Proof of Theorem dedths
StepHypRef Expression
1 dfsbcq 3470 . . 3 (𝑥 = if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) → ([𝑥 / 𝑥]𝜓[if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓))
2 dedths.1 . . . 4 [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓
3 sbcid 3485 . . . . 5 ([𝑥 / 𝑥]𝜑𝜑)
4 ifbi 4140 . . . . 5 (([𝑥 / 𝑥]𝜑𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵))
5 dfsbcq 3470 . . . . 5 (if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵) → ([if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓))
63, 4, 5mp2b 10 . . . 4 ([if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓[if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓)
72, 6mpbir 221 . . 3 [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓
81, 7dedth 4172 . 2 ([𝑥 / 𝑥]𝜑[𝑥 / 𝑥]𝜓)
9 sbcid 3485 . 2 ([𝑥 / 𝑥]𝜓𝜓)
108, 3, 93imtr3i 280 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  [wsbc 3468  ifcif 4119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-sbc 3469  df-if 4120
This theorem is referenced by:  renegclALT  34567
  Copyright terms: Public domain W3C validator