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Theorem dedths 33069
Description: A version of weak deduction theorem dedth 4088 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 3403 . . 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 3418 . . . . 5 ([𝑥 / 𝑥]𝜑𝜑)
4 ifbi 4056 . . . . 5 (([𝑥 / 𝑥]𝜑𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵))
5 dfsbcq 3403 . . . . 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 219 . . 3 [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓
81, 7dedth 4088 . 2 ([𝑥 / 𝑥]𝜑[𝑥 / 𝑥]𝜓)
9 sbcid 3418 . 2 ([𝑥 / 𝑥]𝜓𝜓)
108, 3, 93imtr3i 278 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  [wsbc 3401  ifcif 4035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-sbc 3402  df-if 4036
This theorem is referenced by:  renegclALT  33070
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