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Theorem dedths 36531
 Description: A version of weak deduction theorem dedth 4479 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 3699 . . 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 3714 . . . . 5 ([𝑥 / 𝑥]𝜑𝜑)
4 ifbi 4443 . . . . 5 (([𝑥 / 𝑥]𝜑𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵))
5 dfsbcq 3699 . . . . 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 234 . . 3 [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓
81, 7dedth 4479 . 2 ([𝑥 / 𝑥]𝜑[𝑥 / 𝑥]𝜓)
9 sbcid 3714 . 2 ([𝑥 / 𝑥]𝜓𝜓)
108, 3, 93imtr3i 295 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1539  [wsbc 3697  ifcif 4421 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-sbc 3698  df-if 4422 This theorem is referenced by:  renegclALT  36532
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