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Theorem dedths2 35987
 Description: Generalization of dedths 35984 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
Hypothesis
Ref Expression
dedths2.1 [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓
Assertion
Ref Expression
dedths2 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Proof of Theorem dedths2
StepHypRef Expression
1 dfsbcq 3778 . 2 (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜓[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓))
2 dedths2.1 . 2 [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓
31, 2dedth 4526 1 ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsbc 3776  ifcif 4470 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-sbc 3777  df-if 4471 This theorem is referenced by: (None)
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