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Theorem elimhyps2 34568
Description: Generalization of elimhyps 34565 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 3470 . 2 (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜑[if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑))
2 dfsbcq 3470 . 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 4179 1 [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  [wsbc 3468  ifcif 4119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-sbc 3469  df-if 4120
This theorem is referenced by: (None)
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