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Theorem elimhyps2 35980
Description: Generalization of elimhyps 35977 that is not useful unless we can separately prove 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.)
Hypothesis
Ref Expression
elimhyps2.1 [𝐵 / 𝑥]𝜑
Assertion
Ref Expression
elimhyps2 [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑

Proof of Theorem elimhyps2
StepHypRef Expression
1 dfsbcq 3771 . 2 (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜑[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑))
2 dfsbcq 3771 . 2 (𝐵 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐵 / 𝑥]𝜑[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑))
3 elimhyps2.1 . 2 [𝐵 / 𝑥]𝜑
41, 2, 3elimhyp 4526 1 [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  [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-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: (None)
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