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Theorem dedths 35978
Description: A version of weak deduction theorem dedth 4519 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 3771 . . 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 3786 . . . . 5 ([𝑥 / 𝑥]𝜑𝜑)
4 ifbi 4484 . . . . 5 (([𝑥 / 𝑥]𝜑𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵))
5 dfsbcq 3771 . . . . 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 232 . . 3 [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓
81, 7dedth 4519 . 2 ([𝑥 / 𝑥]𝜑[𝑥 / 𝑥]𝜓)
9 sbcid 3786 . 2 ([𝑥 / 𝑥]𝜓𝜓)
108, 3, 93imtr3i 292 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  [wsbc 3769  ifcif 4463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-sbc 3770  df-if 4464
This theorem is referenced by:  renegclALT  35979
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